Motivated by recent work on stochastic gradient descent methods, we develop two stochastic variants of greedy algorithms for possibly non-convex optimization problems with sparsity constraints. We prove linear convergence in expectation to the solution within a specified tolerance. This generalized framework applies to problems such as sparse signal recovery in compressed sensing, low-rank matrix recovery, and covariance matrix estimation, giving methods with provable convergence guarantees that often outperform their deterministic counterparts. We also analyze the settings where gradients and projections can only be computed approximately, and prove the methods are robust to these approximations. We include many numerical experiments which align with the theoretical analysis and demonstrate these improvements in several different settings.
We outline the GradMP method here, since it will be used as motivation for the work we propose. GradMP @cite_38 is a generalization of the CoSaMP @cite_29 that solves a wider class of sparse reconstruction problems. Like OMP, these methods consist of four main steps: i) form a signal proxy, ii) select a set of large entries of the proxy, iii) use those as the support estimation and estimate the signal via least-squares, and iv) prune the estimation and repeat. Methods like OMP and CoSaMP use the proxy @math ; more general methods like GradMP use the gradient @math (see @cite_38 for details). The analysis of GradMP depends on the restricted strong convexity and restricted strong smoothness properties as in Definitions and below, the first of which is motivated by a similar property introduced in @cite_30 . Under these assumptions, the authors prove linear convergence to the noise floor.
@cite_30 @cite_38 @cite_29
2950190315 2289917018
In this paper, we consider the problem of compressed sensing where the goal is to recover all sparse vectors using a small number of fixed linear measurements. For this problem, we propose a novel partial hard-thresholding operator that leads to a general family of iterative algorithms. While one extreme of the family yields well known hard thresholding algorithms like ITI and HTP[17, 10], the other end of the spectrum leads to a novel algorithm that we call Orthogonal Matching Pursuit with Replacement (OMPR). OMPR, like the classic greedy algorithm OMP, adds exactly one coordinate to the support at each iteration, based on the correlation with the current residual. However, unlike OMP, OMPR also removes one coordinate from the support. This simple change allows us to prove that OMPR has the best known guarantees for sparse recovery in terms of the Restricted Isometry Property (a condition on the measurement matrix). In contrast, OMP is known to have very weak performance guarantees under RIP. Given its simple structure, we are able to extend OMPR using locality sensitive hashing to get OMPR-Hash, the first provably sub-linear (in dimensionality) algorithm for sparse recovery. Our proof techniques are novel and flexible enough to also permit the tightest known analysis of popular iterative algorithms such as CoSaMP and Subspace Pursuit. We provide experimental results on large problems providing recovery for vectors of size up to million dimensions. We demonstrate that for large-scale problems our proposed methods are more robust and faster than existing methods. Orthogonal matching pursuit (OMP) is a commonly used algorithm for recovery sparse signals due to its low complexity and simple implementation. We analyze the convergence property of OMP based on the restricted isometry property (RIP), and show that the OMP algorithm can exactly recover an arbitrary K-sparse signal using K steps provided that the sampling matrix Φ satisfies the RIP with parameter . In addition, we also give the convergence analysis of OMP for the case of inaccurate measurements. Moreover, a variant of OMP, referred to as multi-candidate OMP (MOMP) algorithm, is proposed to recover sparse signals, which can further reduce the computational complexity of OMP. The key point of MOMP is that at each step it selects multi-candidates adding to the optimal atom set, whilst OMP only selects one atom. We also present the convergence analysis of MOMP using the RIP. Finally, we testify the performance of the proposed algorithm using several numerical experiments. Abstract It is now well-known that one can reconstruct sparse or compressible signals accurately from a very limited number of measurements, possibly contaminated with noise. This technique known as “compressed sensing” or “compressive sampling” relies on properties of the sensing matrix such as the restricted isometry property . In this Note, we establish new results about the accuracy of the reconstruction from undersampled measurements which improve on earlier estimates, and have the advantage of being more elegant. To cite this article: E.J. Candes, C. R. Acad. Sci. Paris, Ser. I 346 (2008). Hybrid multiple input multiple output (MIMO) systems consist of an analog beamformer with large antenna arrays followed by a digital MIMO processor. Channel estimation for hybrid MIMO systems in millimeter wave (mm-wave) communications is challenging because of the large antenna array and the low signal-to-noise ratio (SNR) before beamforming. In this paper, we propose an open-loop channel estimator for mm-wave hybrid MIMO systems exploiting the sparse nature of mm-wave channels. A sparse signal recovery problem is formulated for channel estimation and solved by the orthogonal matching pursuit (OMP) based methods. A modification of the OMP algorithm, called the multi-grid (MG) OMP, is proposed. It is shown that the MG-OMP can significantly reduce the computational load of the OMP method. A process for designing the training beams is also developed. Specifically, given the analog training beams the baseband processor for beam training is designed. Simulation results demonstrate the advantage of the OMP based methods over the conventional least squares (LS) method and the efficiency of the MG-OMP over the original OMP. In recent years scholars have built maps of science by connecting the academic fields that cite each other, are cited together, or that cite a similar literature. But since scholars cannot always publish in the fields they cite, or that cite them, these science maps are only rough proxies for the potential of a scholar, organization, or country, to enter a new academic field. Here we use a large dataset of scholarly publications disambiguated at the individual level to create a map of science-or research space-where links connect pairs of fields based on the probability that an individual has published in both of them. We find that the research space is a significantly more accurate predictor of the fields that individuals and organizations will enter in the future than citation based science maps. At the country level, however, the research space and citations based science maps are equally accurate. These findings show that data on career trajectories-the set of fields that individuals have previously published in-provide more accurate predictors of future research output for more focalized units-such as individuals or organizations-than citation based science maps. Orthogonal matching pursuit (OMP) algorithm is an efficient method for the recovery of a sparse signal in compressed sensing, due to its ease implementation and low complexity. In this paper, the robustness of the OMP algorithm under the restricted isometry property (RIP) is presented. It is shown that ( _K + K _ K,1 < 1 ) is sufficient for the OMP algorithm to recover exactly the support of arbitrary K-sparse signal if its nonzero components are large enough for both l 2 bounded and l ∞ bounded noises. One of the mysteries in the success of neural networks is randomly initialized first order methods like gradient descent can achieve zero training loss even though the objective function is non-convex and non-smooth. This paper demystifies this surprising phenomenon for two-layer fully connected ReLU activated neural networks. For an @math hidden node shallow neural network with ReLU activation and @math training data, we show as long as @math is large enough and no two inputs are parallel, randomly initialized gradient descent converges to a globally optimal solution at a linear convergence rate for the quadratic loss function. Our analysis relies on the following observation: over-parameterization and random initialization jointly restrict every weight vector to be close to its initialization for all iterations, which allows us to exploit a strong convexity-like property to show that gradient descent converges at a global linear rate to the global optimum. We believe these insights are also useful in analyzing deep models and other first order methods. As a paradigm for reconstructing sparse signals using a set of under sampled measurements, compressed sensing has received much attention in recent years. In identifying the sufficient condition under which the perfect recovery of sparse signals is ensured, a property of the sensing matrix referred to as the restricted isometry property (RIP) is popularly employed. In this article, we propose the RIP based bound of the orthogonal matching pursuit (OMP) algorithm guaranteeing the exact reconstruction of sparse signals. Our proof is built on an observation that the general step of the OMP process is in essence the same as the initial step in the sense that the residual is considered as a new measurement preserving the sparsity level of an input vector. Our main conclusion is that if the restricted isometry constant δ K of the sensing matrix satisfies δ K < K - 1 K - 1 + K then the OMP algorithm can perfectly recover K(> 1)-sparse signals from measurements. We show that our bound is sharp and indeed close to the limit conjectured by Dai and Milenkovic. We shall show that if the restricted isometry constant (RIC) @math of the measurement matrix @math satisfies @math then the greedy algorithm Orthogonal Matching Pursuit(OMP) will succeed. That is, OMP can recover every @math -sparse signal @math in @math iterations from @math . Moreover, we shall show the upper bound of RIC is sharp in the following sense. For any given @math , we shall construct a matrix @math with the RIC @math such that OMP may not recover some @math -sparse signal @math in @math iterations. Orthogonal matching pursuit (OMP) is the canonical greedy algorithm for sparse approximation. In this paper we demonstrate that the restricted isometry property (RIP) can be used for a very straightforward analysis of OMP. Our main conclusion is that the RIP of order K+1 (with isometry constant δ <; [ 1 ( 3√K)]) is sufficient for OMP to exactly recover any K-sparse signal. The analysis relies on simple and intuitive observations about OMP and matrices which satisfy the RIP. For restricted classes of K-sparse signals (those that are highly compressible), a relaxed bound on the isometry constant is also established. A deeper understanding of OMP may benefit the analysis of greedy algorithms in general. To demonstrate this, we also briefly revisit the analysis of the regularized OMP (ROMP) algorithm. Compressive sampling (CoSa) is a new methodology which demonstrates that sparse signals can be recovered from a small number of linear measurements. Greedy algorithms like CoSaMP have been designed for this recovery, and variants of these methods have been adapted to the case where sparsity is with respect to some arbitrary dictionary rather than an orthonormal basis. In this work we present an analysis of the so-called Signal Space CoSaMP method when the measurements are corrupted with mean-zero white Gaussian noise. We establish near-oracle performance for recovery of signals sparse in some arbitrary dictionary. In addition, we analyze the block variant of the method for signals whose supports obey a block structure, extending the method into the model-based compressed sensing framework. Numerical experiments confirm that the block method significantly outperforms the standard method in these settings. We study the minimization of a convex function @math over the set of @math positive semi-definite matrices, but when the problem is recast as @math , with @math and @math . We study the performance of gradient descent on @math ---which we refer to as Factored Gradient Descent (FGD)---under standard assumptions on the original function @math . We provide a rule for selecting the step size and, with this choice, show that the local convergence rate of FGD mirrors that of standard gradient descent on the original @math : i.e., after @math steps, the error is @math for smooth @math , and exponentially small in @math when @math is (restricted) strongly convex. In addition, we provide a procedure to initialize FGD for (restricted) strongly convex objectives and when one only has access to @math via a first-order oracle; for several problem instances, such proper initialization leads to global convergence guarantees. FGD and similar procedures are widely used in practice for problems that can be posed as matrix factorization. To the best of our knowledge, this is the first paper to provide precise convergence rate guarantees for general convex functions under standard convex assumptions. This paper studies sparse spikes deconvolution over the space of measures. We focus on the recovery properties of the support of the measure (i.e., the location of the Dirac masses) using total variation of measures (TV) regularization. This regularization is the natural extension of the @math l1 norm of vectors to the setting of measures. We show that support identification is governed by a specific solution of the dual problem (a so-called dual certificate) having minimum @math L2 norm. Our main result shows that if this certificate is non-degenerate (see the definition below), when the signal-to-noise ratio is large enough TV regularization recovers the exact same number of Diracs. We show that both the locations and the amplitudes of these Diracs converge toward those of the input measure when the noise drops to zero. Moreover, the non-degeneracy of this certificate can be checked by computing a so-called vanishing derivative pre-certificate. This proxy can be computed in closed form by solving a linear system. Lastly, we draw connections between the support of the recovered measure on a continuous domain and on a discretized grid. We show that when the signal-to-noise level is large enough, and provided the aforementioned dual certificate is non-degenerate, the solution of the discretized problem is supported on pairs of Diracs which are neighbors of the Diracs of the input measure. This gives a precise description of the convergence of the solution of the discretized problem toward the solution of the continuous grid-free problem, as the grid size tends to zero. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We consider the problem of estimating an unknown signal @math from noisy linear observations @math . In many practical instances, @math has a certain structure that can be captured by a structure inducing convex function @math . For example, @math norm can be used to encourage a sparse solution. To estimate @math with the aid of @math , we consider the well-known LASSO method and provide sharp characterization of its performance. We assume the entries of the measurement matrix @math and the noise vector @math have zero-mean normal distributions with variances @math and @math respectively. For the LASSO estimator @math , we attempt to calculate the Normalized Square Error (NSE) defined as @math as a function of the noise level @math , the number of observations @math and the structure of the signal. We show that, the structure of the signal @math and choice of the function @math enter the error formulae through the summary parameters @math and @math , which are defined as the Gaussian squared-distances to the subdifferential cone and to the @math -scaled subdifferential, respectively. The first LASSO estimator assumes a-priori knowledge of @math and is given by @math . We prove that its worst case NSE is achieved when @math and concentrates around @math . Secondly, we consider @math , for some @math . This time the NSE formula depends on the choice of @math and is given by @math . We then establish a mapping between this and the third estimator @math . Finally, for a number of important structured signal classes, we translate our abstract formulae to closed-form upper bounds on the NSE. Applications of compressed sensing motivate the possibility of using different operators to encode and decode a signal of interest. Since it is clear that the operators cannot be too different, we can view the discrepancy between the two matrices as a perturbation. The stability of l 1 -minimization and greedy algorithms to recover the signal in the presence of additive noise is by now well-known. Recently however, work has been done to analyze these methods with noise in the measurement matrix, which generates a multiplicative noise term. This new framework of generalized perturbations (i.e., both additive and multiplicative noise) extends the prior work on stable signal recovery from incomplete and inaccurate measurements of Candes, Romberg and Tao using Basis Pursuit (BP), and of Needell and Tropp using Compressive Sampling Matching Pursuit (CoSaMP). We show, under reasonable assumptions, that the stability of the reconstructed signal by both BP and CoSaMP is limited by the noise level in the observation. Our analysis extends easily to arbitrary greedy methods. This paper presents a new analysis for the orthogonal matching pursuit (OMP) algorithm. It is shown that if the restricted isometry property (RIP) is satisfied at sparsity level O(k), then OMP can stably recover a k-sparse signal in 2-norm under measurement noise. For compressed sensing applications, this result implies that in order to uniformly recover a k-sparse signal in Rd, only O(k lnd) random projections are needed. This analysis improves some earlier results on OMP depending on stronger conditions that can only be satisfied with Ω(k2 lnd) or Ω(k1.6 lnd) random projections. Approximate message passing (AMP) is a class of efficient algorithms for solving high-dimensional linear regression tasks where one wishes to recover an unknown signal from noisy, linear measurements y = A + w. When applying a separable denoiser at each iteration, the performance of AMP (for example, the mean squared error of its estimates) can be accurately tracked by a simple, scalar iteration referred to as state evolution. Although separable denoisers are sufficient if the unknown signal has independent and identically distributed entries, in many real-world applications, like image or audio signal reconstruction, the unknown signal contains dependencies between entries. In these cases, a coordinate-wise independence structure is not a good approximation to the true prior of the unknown signal. In this paper we assume the unknown signal has dependent entries, and using a class of non-separable sliding-window denoisers, we prove that a new form of state evolution still accurately predicts AMP performance. This is an early step in understanding the role of non-separable denoisers within AMP, and will lead to a characterization of more general denoisers in problems including compressive image reconstruction. We obtain an improved finite-sample guarantee on the linear convergence of stochastic gradient descent for smooth and strongly convex objectives, improving from a quadratic dependence on the conditioning @math (where @math is a bound on the smoothness and @math on the strong convexity) to a linear dependence on @math . Furthermore, we show how reweighting the sampling distribution (i.e. importance sampling) is necessary in order to further improve convergence, and obtain a linear dependence in the average smoothness, dominating previous results. We also discuss importance sampling for SGD more broadly and show how it can improve convergence also in other scenarios. Our results are based on a connection we make between SGD and the randomized Kaczmarz algorithm, which allows us to transfer ideas between the separate bodies of literature studying each of the two methods. In particular, we recast the randomized Kaczmarz algorithm as an instance of SGD, and apply our results to prove its exponential convergence, but to the solution of a weighted least squares problem rather than the original least squares problem. We then present a modified Kaczmarz algorithm with partially biased sampling which does converge to the original least squares solution with the same exponential convergence rate. Approximate message passing (AMP) is a class of efficient algorithms for solving high-dimensional linear regression tasks where one wishes to recover an unknown signal βο from noisy, linear measurements y = Αβ 0 + w. When applying a separable denoiser at each iteration, the performance of AMP (for example, the mean squared error of its estimates) can be accurately tracked by a simple, scalar iteration referred to as state evolution. Although separable denoisers are sufficient if the unknown signal has independent and identically distributed entries, in many real-world applications, like image or audio signal reconstruction, the unknown signal contains dependencies between entries. In these cases, a coordinate-wise independence structure is not a good approximation to the true prior of the unknown signal. In this paper we assume the unknown signal has dependent entries, and using a class of non-separable sliding-window denoisers, we prove that a new form of state evolution still accurately predicts AMP performance. This is an early step in understanding the role of non-separable denoisers within AMP, and will lead to a characterization of more general denoisers in problems including compressive image reconstruction.