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The Recently proposed Vector Approximate Message Passing (VAMP) algorithm demonstrates a great reconstruction potential at solving compressed sensing related linear inverse problems. VAMP provides high per-iteration improvement, can utilize powerful denoisers like BM3D, has rigorously defined dynamics and is able to recover signals measured by highly undersampled and ill-conditioned linear operators. Yet, its applicability is limited to relatively small problem sizes due to the necessity to compute the expensive LMMSE estimator at each iteration. In this work we consider the problem of upscaling VAMP by utilizing Conjugate Gradient (CG) to approximate the intractable LMMSE estimator. We propose a rigorous method for correcting and tuning CG withing CG-VAMP to achieve a stable and efficient reconstruction. To further improve the performance of CG-VAMP, we design a warm-starting scheme for CG and develop theoretical models for the Onsager correction and the State Evolution of Warm-Started CG-VAMP (WS-CG-VAMP). Additionally, we develop robust and accurate methods for implementing the WS-CG-VAMP algorithm. The numerical experiments on large-scale image reconstruction problems demonstrate that WS-CG-VAMP requires much fewer CG iterations compared to CG-VAMP to achieve the same or superior level of reconstruction. [ABSTRACT FROM AUTHOR]

The Binary Iterative Hard Thresholding (BIHT) algorithm is a popular reconstruction method for one-bit compressed sensing due to its simplicity and fast empirical convergence. Despite considerable research on this algorithm, a theoretical understanding of the corresponding approximation error and convergence rate still remains an open problem. This paper shows that the normalized version of BIHT (NBIHT) achieves an approximation error rate optimal up to logarithmic factors. More precisely, using $m$ one-bit measurements of an $s$ -sparse vector $x$ , we prove that the approximation error of NBIHT is of order $O \left ({\frac{1 }{ m }}\right)$ up to logarithmic factors, which matches the information-theoretic lower bound $\Omega \left ({\frac{1 }{ m }}\right)$ proved by Jacques, Laska, Boufounos, and Baraniuk in 2013. To our knowledge, this is the first theoretical analysis of a BIHT-type algorithm that explains the optimal rate of error decay empirically observed in the literature. This also makes NBIHT the first provable computationally-efficient one-bit compressed sensing algorithm that breaks the inverse square-root error decay rate $O \left ({\frac{1 }{ m^{1/2} }}\right)\vphantom {{\left ({\frac{1 }{ m^{1/2} }}\right)}^{'}}$. [ABSTRACT FROM AUTHOR]

This paper proposes Bayes-optimal convolutional approximate message-passing (CAMP) for signal recovery in compressed sensing. CAMP uses the same low-complexity matched filter (MF) for interference suppression as approximate message-passing (AMP). To improve the convergence property of AMP for ill-conditioned sensing matrices, the so-called Onsager correction term in AMP is replaced by a convolution of all preceding messages. The tap coefficients in the convolution are determined so as to realize asymptotic Gaussianity of estimation errors via state evolution (SE) under the assumption of orthogonally invariant sensing matrices. An SE equation is derived to optimize the sequence of denoisers in CAMP. The optimized CAMP is proved to be Bayes-optimal for all orthogonally invariant sensing matrices if the SE equation converges to a fixed-point and if the fixed-point is unique. For sensing matrices with low-to-moderate condition numbers, CAMP can achieve the same performance as high-complexity orthogonal/vector AMP that requires the linear minimum mean-square error (LMMSE) filter instead of the MF. [ABSTRACT FROM AUTHOR]

We investigate the statistical and algorithmic properties of random neural-network generative priors in a simple inference problem: spiked-matrix estimation. We establish a rigorous expression for the performance of the Bayes-optimal estimator in the high-dimensional regime, and identify the statistical threshold for weak-recovery of the spike. Next, we derive a message-passing algorithm taking into account the latent structure of the spike, and show that its performance is asymptotically optimal for natural choices of the generative network architecture. The absence of an algorithmic gap in this case is in stark contrast to known results for sparse spikes, another popular prior for modelling low-dimensional signals, and for which no algorithm is known to achieve the optimal statistical threshold. Finally, we show that linearising our message passing algorithm yields a simple spectral method also achieving the optimal threshold for reconstruction. We conclude with an experiment on a real data set showing that our bespoke spectral method outperforms vanilla PCA. [ABSTRACT FROM AUTHOR]

In the problem of compressed phase retrieval, the goal is to reconstruct a sparse or approximately $k$ -sparse vector $x \in \mathbb {C} ^{n}$ given access to $y= |\Phi x|$ , where $|v|$ denotes the vector obtained from taking the absolute value of $v\in \mathbb {C} ^{n}$ coordinate-wise. In this paper we present sublinear-time algorithms for a few for-each variants of the compressive phase retrieval problem which are akin to the variants considered for the classical compressive sensing problem in theoretical computer science. Our algorithms use pure combinatorial techniques and near-optimal number of measurements. [ABSTRACT FROM AUTHOR]

Signal recovery from unitarily invariant measurements is investigated in this paper. A message-passing algorithm is formulated on the basis of expectation propagation (EP). A rigorous analysis is presented for the dynamics of the algorithm in the large system limit, where both input and output dimensions tend to infinity while the compression rate is kept constant. The main result is the justification of state evolution (SE) equations conjectured by Ma and Ping. This result implies that the EP-based algorithm achieves the Bayes-optimal performance that was originally derived via a non-rigorous tool in statistical physics and proved partially in a recent paper, when the compression rate is larger than a threshold. The proof is based on an extension of a conventional conditioning technique for the standard Gaussian matrix to the case of the Haar matrix. [ABSTRACT FROM AUTHOR]

We consider the problem of decoding a discrete signal of categorical variables from the observation of several histograms of pooled subsets of it. We present an approximate message passing (AMP) algorithm for recovering the signal in the random dense setting where each observed histogram involves a random subset of entries of size proportional to $n$. We characterize the performance of the algorithm in the asymptotic regime where the number of observations $m$ tends infinity proportionally to $n$ by deriving the corresponding state evolution (SE) equations and studying their dynamics. We initiate the analysis of the multi-dimensional SE dynamics by proving their convergence to a fixed point, along with some further properties of the iterates. The analysis reveals sharp phase transition phenomena where the behavior of AMP changes from exact recovery to weak correlation with the signal, as $m/n$ crosses a threshold. We derive formulae for the threshold in some special cases and show that they accurately match experimental behavior. [ABSTRACT FROM AUTHOR]

Approximate message passing (AMP) refers to a class of efficient algorithms for statistical estimation in high-dimensional problems such as compressed sensing and low-rank matrix estimation. This paper analyzes the performance of AMP in the regime where the problem dimension is large but finite. For concreteness, we consider the setting of high-dimensional regression, where the goal is to estimate a high-dimensional vector $\beta _{0}$ from a noisy measurement $y=A \beta _{0} + w$. AMP is a low-complexity, scalable algorithm for this problem. Under suitable assumptions on the measurement matrix $A$ , AMP has the attractive feature that its performance can be accurately characterized in the large system limit by a simple scalar iteration called state evolution. Previous proofs of the validity of state evolution have all been asymptotic convergence results. In this paper, we derive a concentration inequality for AMP with Gaussian matrices with independent and identically distributed (i.i.d.) entries and finite dimension $n \times N$. The result shows that the probability of deviation from the state evolution prediction falls exponentially in $n$. This provides theoretical support for empirical findings that have demonstrated excellent agreement of AMP performance with state evolution predictions for moderately large dimensions. The concentration inequality also indicates that the number of AMP iterations $t$ can grow no faster than order $({\log n}/{\log \log n})$ for the performance to be close to the state evolution predictions with high probability. The analysis can be extended to obtain similar non-asymptotic results for AMP in other settings such as low-rank matrix estimation. [ABSTRACT FROM AUTHOR]

We study the convergence of the iterative projected gradient (IPG) algorithm for arbitrary (possibly non-convex) sets when both the gradient and projection oracles are computed approximately. We consider different notions of approximation of which we show that the progressive fixed precision and the $(1+ \varepsilon)$ -optimal oracles can achieve the same accuracy as for the exact IPG algorithm. We show that the former scheme is also able to maintain the (linear) rate of convergence of the exact algorithm under the same embedding assumption. In contrast, the $(1+ \varepsilon)$ -approximate oracle requires a stronger embedding condition, moderate compression ratios and it typically slows down the convergence. We apply our results to accelerate solving a class of data driven compressed sensing problems, where we replace iterative exhaustive searches over large data sets by fast approximate nearest neighbor search strategies based on the cover tree data structure. For data sets with low intrinsic dimensions, our proposed algorithm achieves a complexity logarithmic in terms of the data set population as opposed to the linear complexity of a brute force search. By running several numerical experiments, we conclude similar observations as predicted by our theoretical analysis. [ABSTRACT FROM AUTHOR]

Sparsity-based subspace clustering algorithms have attracted significant attention thanks to their excellent performance in practical applications. A prominent example is the sparse subspace clustering (SSC) algorithm by Elhamifar and Vidal, which performs spectral clustering based on an adjacency matrix obtained by sparsely representing each data point in terms of all the other data points via the Lasso. When the number of data points is large or the dimension of the ambient space is high, the computational complexity of SSC quickly becomes prohibitive. Dyer et al. observed that SSC-orthogonal matching pursuit (OMP) obtained by replacing the Lasso by the greedy OMP algorithm results in significantly lower computational complexity, while often yielding comparable performance. The central goal of this paper is an analytical performance characterization of SSC-OMP for noisy data. Moreover, we introduce and analyze the SSC-matching pursuit (MP) algorithm, which employs MP in lieu of OMP. Both SSC-OMP and SSC-MP are proven to succeed even when the subspaces intersect and when the data points are contaminated by severe noise. The clustering conditions we obtain for SSC-OMP and SSC-MP are similar to those for SSC and for the thresholding-based subspace clustering (TSC) algorithm due to Heckel and Bölcskei. Analytical results in combination with numerical results indicate that both SSC-OMP and SSC-MP with a data-dependent stopping criterion automatically detect the dimensions of the subspaces underlying the data. Experiments on synthetic and on real data show that SSC-MP often matches or exceeds the performance of the computationally more expensive SSC-OMP algorithm. Moreover, SSC-MP compares very favorably to SSC, TSC, and the nearest subspace neighbor algorithm, both in terms of clustering performance and running time. In addition, we find that, in contrast to SSC-OMP, the performance of SSC-MP is very robust with respect to the choice of parameters in the stopping criteria. [ABSTRACT FROM PUBLISHER]

In many application areas ranging from bioinformatics to imaging, we are faced with the following question: can we recover a sparse vector xo \in \mathbb {R}^{N} from its undersampled set of noisy observations y \in \mathbb R^n , y= A xo+ w . The last decade has witnessed a surge of algorithms and theoretical results to address this question. One of the most popular schemes is the \ell p -regularized least squares given by the following formulation: \hat x(\gamma ,p ) \in \arg \min x~({1}/{2})\| {y - Ax} \|2^{2} + \gamma {\| x \|p^{p}} , where p \in [0, 1] . Among these optimization problems, the case p = 1 , also known as LASSO, is the best accepted in practice, for the following two reasons. First, thanks to the extensive studies performed in the fields of high-dimensional statistics and compressed sensing, we have a clear picture of LASSO’s performance. Second, it is convex and efficient algorithms exist for finding its global minima. Unfortunately, neither of the above two properties hold for 0 \leq p<1 . However, they are still appealing because of the following folklores in the high-dimensional statistics. First, \hat x(\gamma , p ) is closer to than \hat {x}(\gamma ,1) . Second, if we employ iterative methods that aim to converge to a local minima of {\arg \min }_{x}~({1}/{2})\| {y - Ax} \|_{2}^{2} + \gamma {\| x \|_{p}^{p}} , then under good initialization, these algorithms converge to a solution that is still closer to xo than \hat {x}(\gamma ,1) . In spite of the existence of plenty of empirical results that support these folklore theorems, the theoretical progress to establish them has been very limited. This paper aims to study the above-mentioned folklore theorems and establish their scope of validity. Starting with approximate message passing (AMP) algorithm as a heuristic method for solving \ell p -regularized least squares, we study the following questions. First, what is the impact of initialization on the performance of the algorithm? Second, when does the algorithm recover the sparse signal xo under a “good” initialization? Third, when does the algorithm converge to the sparse signal regardless of the initialization? Studying these questions will not only shed light on the second folklore theorem, but also lead us to the answer the first one, i.e., the performance of the global optima \hat x(\gamma , p ) . For that purpose, we employ the replica analysis1 to show the connection between the solution of AMP and \hat {x}(\gamma , p) in the asymptotic settings. This enables us to compare the accuracy of \hat x(\gamma ,p )$ and $\hat x(\gamma ,1 )$ . In particular, we will present an accurate characterization of the phase transition and noise sensitivity of \ell p -regularized least squares for every $0 \leq p \leq 1$ . Our results in the noiseless setting confirm that \ell p -regularized least squares (if $\gamma $ is tuned optimally) exhibits the same phase transition for every $0 \leq p< 1$ and this phase transition is much better than that of LASSO. Furthermore, we show that in the noisy setting, there is a major difference between the performance of \ell p -regularized least squares with different values of $p$ . For instance, we will show that for very small and very large measurement noises, $p = 0$ and $p = 1$ outperform the other values of $p$ , respectively.1

Replica method is a widely accepted heuristic in statistical physics for analyzing large disordered systems.

As shown by Blumensath and Davies (2009) and Baraniuk et al. (2010), signals whose wavelet coefficients exhibit a rooted tree structure can be recovered using specially adapted compressed sensing algorithms from just n=\mathcal O(k) measurements, where $k$ is the sparsity of the signal. Motivated by these results, we introduce a simplified proportional-dimensional asymptotic framework, which enables the quantitative evaluation of recovery guarantees for tree-based compressed sensing. In the context of Gaussian matrices, we apply this framework to existing worst-case analysis of the iterative tree projection (ITP) algorithm, which makes use of the tree-based restricted isometry property (RIP). Within the same framework, we then obtain quantitative results based on a new method of analysis, which considers the fixed points of the algorithm. By exploiting the realistic average-case assumption that the measurements are statistically independent of the signal, we obtain significant quantitative improvements when compared with the tree-based RIP analysis. Our results have a refreshingly simple interpretation, explicitly determining a bound on the number of measurements that are required as a multiple of the sparsity. For example, we prove that exact recovery of binary tree-based signals from noiseless Gaussian measurements is asymptotically guaranteed for ITP with constant stepsize provided $n\geq 50k$ . All our results extend to the more realistic case in which measurements are corrupted by noise. [ABSTRACT FROM PUBLISHER]

The estimation of a random vector with independent components passed through a linear transform followed by a componentwise (possibly nonlinear) output map arises in a range of applications. Approximate message passing (AMP) methods, based on Gaussian approximations of loopy belief propagation, have recently attracted considerable attention for such problems. For large random transforms, these methods exhibit fast convergence and admit precise analytic characterizations with testable conditions for optimality, even for certain non-convex problem instances. However, the behavior of AMP under general transforms is not fully understood. In this paper, we consider the generalized AMP (GAMP) algorithm and relate the method to more common optimization techniques. This analysis enables a precise characterization of the GAMP algorithm fixed points that applies to arbitrary transforms. In particular, we show that the fixed points of the so-called max-sum GAMP algorithm for MAP estimation are critical points of a constrained maximization of the posterior density. The fixed points of the sum-product GAMP algorithm for estimation of the posterior marginals can be interpreted as critical points of a certain free energy. [ABSTRACT FROM PUBLISHER]

Group-based sparsity models are instrumental in linear and non-linear regression problems. The main premise of these models is the recovery of “interpretable” signals through the identification of their constituent groups, which can also provably translate in substantial savings in the number of measurements for linear models in compressive sensing. In this paper, we establish a combinatorial framework for group-model selection problems and highlight the underlying tractability issues. In particular, we show that the group-model selection problem is equivalent to the well-known NP-hard weighted maximum coverage problem. Leveraging a graph-based understanding of group models, we describe group structures that enable correct model selection in polynomial time via dynamic programming. Furthermore, we show that popular group structures can be explained by linear inequalities involving totally unimodular matrices, which afford other polynomial time algorithms based on relaxations. We also present a generalization of the group model that allows for within group sparsity, which can be used to model hierarchical sparsity. Finally, we study the Pareto frontier between approximation error and sparsity budget of group-sparse approximations for two tractable models, among which the tree sparsity model, and illustrate selection and computation tradeoffs between our framework and the existing convex relaxations. [ABSTRACT FROM PUBLISHER]

A denoising algorithm seeks to remove noise, errors, or perturbations from a signal. Extensive research has been devoted to this arena over the last several decades, and as a result, todays denoisers can effectively remove large amounts of additive white Gaussian noise. A compressed sensing (CS) reconstruction algorithm seeks to recover a structured signal acquired using a small number of randomized measurements. Typical CS reconstruction algorithms can be cast as iteratively estimating a signal from a perturbed observation. This paper answers a natural question: How can one effectively employ a generic denoiser in a CS reconstruction algorithm? In response, we develop an extension of the approximate message passing (AMP) framework, called denoising-based AMP (D-AMP), that can integrate a wide class of denoisers within its iterations. We demonstrate that, when used with a high-performance denoiser for natural images, D-AMP offers the state-of-the-art CS recovery performance while operating tens of times faster than competing methods. We explain the exceptional performance of D-AMP by analyzing some of its theoretical features. A key element in D-AMP is the use of an appropriate Onsager correction term in its iterations, which coerces the signal perturbation at each iteration to be very close to the white Gaussian noise that denoisers are typically designed to remove. [ABSTRACT FROM PUBLISHER]

Compressive sensing (CS) states that a sparse signal can be recovered from a small number of linear measurements, and that this recovery can be performed efficiently in polynomial time. The framework of model-based CS (model-CS) leverages additional structure in the signal and provides new recovery schemes that can reduce the number of measurements even further. This idea has led to measurement-efficient recovery schemes for a variety of signal models. However, for any given model, model-CS requires an algorithm that solves the model-projection problem: given a query signal, report the signal in the model that is closest to the query signal. Often, this optimization can be computationally very expensive. Moreover, an approximation algorithm is not sufficient for this optimization to provably succeed. As a result, the model-projection problem poses a fundamental obstacle for extending model-CS to many interesting classes of models. In this paper, we introduce a new framework that we call approximation-tolerant model-CS. This framework includes a range of algorithms for sparse recovery that require only approximate solutions for the model-projection problem. In essence, our work removes the aforementioned obstacle to model-CS, thereby extending model-CS to a much wider class of signal models. Interestingly, all our algorithms involve both the minimization and the maximization variants of the model-projection problem. We instantiate this new framework for a new signal model that we call the constrained earth mover distance (CEMD) model. This model is particularly useful for signal ensembles, where the positions of the nonzero coefficients do not change significantly as a function of spatial (or temporal) location. We develop novel approximation algorithms for both the maximization and the minimization versions of the model-projection problem via graph optimization techniques. Leveraging these algorithms and our framework results in a nearly sample-optimal sparse recovery scheme for the CEMD model. [ABSTRACT FROM AUTHOR]

It is well known that \ell 1 minimization can be used to recover sufficiently sparse unknown signals from compressed linear measurements. Exact thresholds on the sparsity, as a function of the ratio between the system dimensions, so that with high probability almost all sparse signals can be recovered from independent identically distributed (i.i.d.) Gaussian measurements, have been computed and are referred to as weak thresholds. In this paper, we introduce a reweighted \ell 1 recovery algorithm composed of two steps: 1) a standard \ell 1 minimization step to identify a set of entries where the signal is likely to reside and 2) a weighted \ell 1 minimization step where entries outside this set are penalized. For signals where the non-sparse component entries are independent and identically drawn from certain classes of distributions, (including most well-known continuous distributions), we prove a strict improvement in the weak recovery threshold. Our analysis suggests that the level of improvement in the weak threshold depends on the behavior of the distribution at the origin. Numerical simulations verify the distribution dependence of the threshold improvement very well, and suggest that in the case of i.i.d. Gaussian nonzero entries, the improvement can be quite impressive—over 20% in the example we consider. [ABSTRACT FROM AUTHOR]

Compressed sensing is a technique for finding sparse solutions to underdetermined linear systems. This technique relies on properties of the sensing matrix such as the restricted isometry property. Sensing matrices that satisfy this property with optimal parameters are mainly obtained via probabilistic arguments. Deciding whether a given matrix satisfies the restricted isometry property is a nontrivial computational problem. Indeed, it is shown in this paper that restricted isometry parameters cannot be approximated in polynomial time within any constant factor under the assumption that the hidden clique problem is hard. In addition, on the positive side, an improvement on the brute-force enumeration algorithm for checking the restricted isometry property is proposed. [ABSTRACT FROM PUBLISHER]

The nascent field of compressed sensing is founded on the fact that high-dimensional signals with simple structure can be recovered accurately from just a small number of randomized samples. Several specific kinds of structures have been explored in the literature, from sparsity and group sparsity to low-rankness. However, two fundamental questions have been left unanswered. What are the general abstract meanings of structure and simplicity? Do there exist universal algorithms for recovering such simple structured objects from fewer samples than their ambient dimension? In this paper, we address these two questions. Using algorithmic information theory tools such as the Kolmogorov complexity, we provide a unified definition of structure and simplicity. Leveraging this new definition, we develop and analyze an abstract algorithm for signal recovery motivated by Occam's Razor. Minimum complexity pursuit (MCP) requires approximately 2\kappa randomized samples to recover a signal of complexity \kappa and ambient dimension n. We also discuss the performance of the MCP in the presence of measurement noise and with approximately simple signals. [ABSTRACT FROM PUBLISHER]

Compressive sensing (CS) has recently emerged as a powerful framework for acquiring sparse signals. The bulk of the CS literature has focused on the case where the acquired signal has a sparse or compressible representation in an orthonormal basis. In practice, however, there are many signals that cannot be sparsely represented or approximated using an orthonormal basis, but that do have sparse representations in a redundant dictionary. Standard results in CS can sometimes be extended to handle this case provided that the dictionary is sufficiently incoherent or well conditioned, but these approaches fail to address the case of a truly redundant or overcomplete dictionary. In this paper, we describe a variant of the iterative recovery algorithm CoSaMP for this more challenging setting. We utilize the \mbi D-RIP, a condition on the sensing matrix analogous to the well-known restricted isometry property. In contrast to prior work, the method and analysis are “signal-focused”; that is, they are oriented around recovering the signal rather than its dictionary coefficients. Under the assumption that we have a near-optimal scheme for projecting vectors in signal space onto the model family of candidate sparse signals, we provide provable recovery guarantees. Developing a practical algorithm that can provably compute the required near-optimal projections remains a significant open problem, but we include simulation results using various heuristics that empirically exhibit superior performance to traditional recovery algorithms. [ABSTRACT FROM AUTHOR]

Compressed sensing posits that, within limits, one can undersample a sparse signal and yet reconstruct it accurately. Knowing the precise limits to such undersampling is important both for theory and practice. We present a formula that characterizes the allowed undersampling of generalized sparse objects. The formula applies to approximate message passing (AMP) algorithms for compressed sensing, which are here generalized to employ denoising operators besides the traditional scalar soft thresholding denoiser. This paper gives several examples including scalar denoisers not derived from convex penalization—the firm shrinkage nonlinearity and the minimax nonlinearity—and also nonscalar denoisers—block thresholding, monotone regression, and total variation minimization. Let the variables \varepsilon = k/N and \delta = n/N denote the generalized sparsity and undersampling fractions for sampling the k-generalized-sparse N-vector x0 according to y=Ax0. Here, A is an n\times N measurement matrix whose entries are iid standard Gaussian. The formula states that the phase transition curve \delta = \delta (\varepsilon ) separating successful from unsuccessful reconstruction of x0 by AMP is given by \delta = M(\varepsilon \vert Denoiser) where M(\varepsilon \vert Denoiser) denotes the per-coordinate minimax mean squared error (MSE) of the specified, optimally tuned denoiser in the directly observed problem y = x + z. In short, the phase transition of a noiseless undersampling problem is identical to the minimax MSE in a denoising problem. We prove that this formula follows from state evolution and present numerical results validating it in a wide range of settings. The above formula generates numerous new insights, both in the scalar and in the nonscalar cases. [ABSTRACT FROM PUBLISHER]

Suppose that we observe noisy linear measurements of an unknown signal that can be modeled as the sum of two component signals, each of which arises from a nonlinear submanifold of a high-dimensional ambient space. We introduce successive projections onto incoherent manifolds (SPIN), a first-order projected gradient method to recover the signal components. Despite the nonconvex nature of the recovery problem and the possibility of underdetermined measurements, SPIN provably recovers the signal components, provided that the signal manifolds are incoherent and that the measurement operator satisfies a certain restricted isometry property. SPIN significantly extends the scope of current recovery models and algorithms for low-dimensional linear inverse problems and matches (or exceeds) the current state of the art in terms of performance. [ABSTRACT FROM AUTHOR]

The general theory of greedy approximation is well developed. Much less is known about how specific features of a dictionary can be used to our advantage. In this paper, we discuss incoherent dictionaries. We build a new greedy algorithm which is called the orthogonal super greedy algorithm (OSGA). We show that the rates of convergence of OSGA and the orthogonal matching pursuit (OMP) with respect to incoherent dictionaries are the same. Based on the analysis of the number of orthogonal projections and the number of iterations, we observed that OSGA(s) is s times simpler (more efficient) than OMP. Greedy approximation is also a fundamental tool for sparse signal recovery. The performance of orthogonal multimatching pursuit, a counterpart of OSGA in the compressed sensing setting, is also analyzed under restricted isometry property conditions. [ABSTRACT FROM AUTHOR]

Consider the noisy underdetermined system of linear equations: y=Ax_0 + z, with A an n \times N measurement matrix, n < N, and z \sim \ssr N(0,\sigma^2 I) a Gaussian white noise. Both y and A are known, both x_0 and z are unknown, and we seek an approximation to x_0. When x_0 has few nonzeros, useful approximations are often obtained by \ell_1-penalized \ell_2 minimization, in which the reconstruction \mathhatx^1,\lambda solves \min\\Vert y - Ax\Vert_2^2/2 + \lambda \Vert x\Vert_1\. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of sampling and reconstructing signals which are assumed to lie on or close to one of several subspaces of a Hilbert space. Importantly, we here consider a very general setting in which we allow infinitely many subspaces in infinite dimensional Hilbert spaces. This general approach allows us to unify many results derived recently in areas such as compressed sensing, affine rank minimization, analog compressed sensing and structured matrix decompositions. [ABSTRACT FROM AUTHOR]

“Approximate message passing” (AMP) algorithms have proved to be effective in reconstructing sparse signals from a small number of incoherent linear measurements. Extensive numerical experiments further showed that their dynamics is accurately tracked by a simple one-dimensional iteration termed state evolution. In this paper, we provide rigorous foundation to state evolution. We prove that indeed it holds asymptotically in the large system limit for sensing matrices with independent and identically distributed Gaussian entries. While our focus is on message passing algorithms for compressed sensing, the analysis extends beyond this setting, to a general class of algorithms on dense graphs. In this context, state evolution plays the role that density evolution has for sparse graphs. The proof technique is fundamentally different from the standard approach to density evolution, in that it copes with a large number of short cycles in the underlying factor graph. It relies instead on a conditioning technique recently developed by Erwin Bolthausen in the context of spin glass theory. [ABSTRACT FROM AUTHOR]