Showing papers on "Iterative method published in 2009"

Subspace Pursuit for Compressive Sensing Signal Reconstruction

Abstract: We propose a new method for reconstruction of sparse signals with and without noisy perturbations, termed the subspace pursuit algorithm. The algorithm has two important characteristics: low computational complexity, comparable to that of orthogonal matching pursuit techniques when applied to very sparse signals, and reconstruction accuracy of the same order as that of linear programming (LP) optimization methods. The presented analysis shows that in the noiseless setting, the proposed algorithm can exactly reconstruct arbitrary sparse signals provided that the sensing matrix satisfies the restricted isometry property with a constant parameter. In the noisy setting and in the case that the signal is not exactly sparse, it can be shown that the mean-squared error of the reconstruction is upper-bounded by constant multiples of the measurement and signal perturbation energies.

Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems

Abstract: This paper studies gradient-based schemes for image denoising and deblurring problems based on the discretized total variation (TV) minimization model with constraints. We derive a fast algorithm for the constrained TV-based image deburring problem. To achieve this task, we combine an acceleration of the well known dual approach to the denoising problem with a novel monotone version of a fast iterative shrinkage/thresholding algorithm (FISTA) we have recently introduced. The resulting gradient-based algorithm shares a remarkable simplicity together with a proven global rate of convergence which is significantly better than currently known gradient projections-based methods. Our results are applicable to both the anisotropic and isotropic discretized TV functionals. Initial numerical results demonstrate the viability and efficiency of the proposed algorithms on image deblurring problems with box constraints.

Sparse Reconstruction by Separable Approximation

Abstract: Finding sparse approximate solutions to large underdetermined 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 (lscr 2) error term added to a sparsity-inducing (usually lscr1) 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 convexity of the regularizer), we prove convergence of the proposed iterative algorithm to a minimum of the objective function. In addition to solving the standard lscr2-lscr1 case, our framework yields efficient solution techniques for other regularizers, such as an lscrinfin 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 lscr2-lscr1 problem, as well as being efficient on problems with other separable regularization terms.

Sparse subspace clustering

Abstract: We propose a method based on sparse representation (SR) to cluster data drawn from multiple low-dimensional linear or affine subspaces embedded in a high-dimensional space. Our method is based on the fact that each point in a union of subspaces has a SR with respect to a dictionary formed by all other data points. In general, finding such a SR is NP hard. Our key contribution is to show that, under mild assumptions, the SR can be obtained `exactly' by using l1 optimization. The segmentation of the data is obtained by applying spectral clustering to a similarity matrix built from this SR. Our method can handle noise, outliers as well as missing data. We apply our subspace clustering algorithm to the problem of segmenting multiple motions in video. Experiments on 167 video sequences show that our approach significantly outperforms state-of-the-art methods.

Uniform Uncertainty Principle and Signal Recovery via Regularized Orthogonal Matching Pursuit

Abstract: This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements—L1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of L1-minimization. Our algorithm, ROMP, reconstructs a sparse signal in a number of iterations linear in the sparsity, and the reconstruction is exact provided the linear measurements satisfy the uniform uncertainty principle.

A Randomized Kaczmarz Algorithm with Exponential Convergence

Abstract: The Kaczmarz method for solving linear systems of equations is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Despite the popularity of this method, useful theoretical estimates for its rate of convergence are still scarce. We introduce a randomized version of the Kaczmarz method for consistent, overdetermined linear systems and we prove that it converges with expected exponential rate. Furthermore, this is the first solver whose rate does not depend on the number of equations in the system. The solver does not even need to know the whole system but only a small random part of it. It thus outperforms all previously known methods on general extremely overdetermined systems. Even for moderately overdetermined systems, numerical simulations as well as theoretical analysis reveal that our algorithm can converge faster than the celebrated conjugate gradient algorithm. Furthermore, our theory and numerical simulations confirm a prediction of Feichtinger et al. in the context of reconstructing bandlimited functions from nonuniform sampling.

Adaptive Dynamic Programming: An Introduction

Abstract: In this article, we introduce some recent research trends within the field of adaptive/approximate dynamic programming (ADP), including the variations on the structure of ADP schemes, the development of ADP algorithms and applications of ADP schemes. For ADP algorithms, the point of focus is that iterative algorithms of ADP can be sorted into two classes: one class is the iterative algorithm with initial stable policy; the other is the one without the requirement of initial stable policy. It is generally believed that the latter one has less computation at the cost of missing the guarantee of system stability during iteration process. In addition, many recent papers have provided convergence analysis associated with the algorithms developed. Furthermore, we point out some topics for future studies.

Neural-Network-Based Near-Optimal Control for a Class of Discrete-Time Affine Nonlinear Systems With Control Constraints

Abstract: In this paper, the near-optimal control problem for a class of nonlinear discrete-time systems with control constraints is solved by iterative adaptive dynamic programming algorithm. First, a novel nonquadratic performance functional is introduced to overcome the control constraints, and then an iterative adaptive dynamic programming algorithm is developed to solve the optimal feedback control problem of the original constrained system with convergence analysis. In the present control scheme, there are three neural networks used as parametric structures for facilitating the implementation of the iterative algorithm. Two examples are given to demonstrate the convergence and feasibility of the proposed optimal control scheme.

Convergence Speed in Distributed Consensus and Averaging

Abstract: We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter. We first consider the case of a fixed communication topology. We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm. We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds. We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm.

Development of EMD-Based Denoising Methods Inspired by Wavelet Thresholding

Abstract: One of the tasks for which empirical mode decomposition (EMD) is potentially useful is nonparametric signal denoising, an area for which wavelet thresholding has been the dominant technique for many years. In this paper, the wavelet thresholding principle is used in the decomposition modes resulting from applying EMD to a signal. We show that although a direct application of this principle is not feasible in the EMD case, it can be appropriately adapted by exploiting the special characteristics of the EMD decomposition modes. In the same manner, inspired by the translation invariant wavelet thresholding, a similar technique adapted to EMD is developed, leading to enhanced denoising performance.

Enhanced Karnik--Mendel Algorithms

Abstract: The Karnik-Mendel (KM) algorithms are iterative procedures widely used in fuzzy logic theory. They are known to converge monotonically and superexponentially fast; however, several (usually two to six) iterations are still needed before convergence occurs. Methods to reduce their computational cost are proposed in this paper. Extensive simulations show that, on average, the enhanced KM algorithms can save about two iterations, which corresponds to more than a 39% reduction in computation time. An additional (at least) 23% computational cost can be saved if no sorting of the inputs is needed.

Spectral Curvature Clustering (SCC)

Abstract: This paper presents novel techniques for improving the performance of a multi-way spectral clustering framework (Govindu in Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05), vol. 1, pp. 1150---1157, 2005; Chen and Lerman, 2007, preprint in the supplementary webpage) for segmenting affine subspaces. Specifically, it suggests an iterative sampling procedure to improve the uniform sampling strategy, an automatic scheme of inferring the tuning parameter from the data, a precise initialization procedure for K-means, as well as a simple strategy for isolating outliers. The resulting algorithm, Spectral Curvature Clustering (SCC), requires only linear storage and takes linear running time in the size of the data. It is supported by theory which both justifies its successful performance and guides our practical choices. We compare it with other existing methods on a few artificial instances of affine subspaces. Application of the algorithm to several real-world problems is also discussed.

Neural network approach to continuous-time direct adaptive optimal control for partially unknown nonlinear systems

Abstract: In this paper we present in a continuous-time framework an online approach to direct adaptive optimal control with infinite horizon cost for nonlinear systems. The algorithm converges online to the optimal control solution without knowledge of the internal system dynamics. Closed-loop dynamic stability is guaranteed throughout. The algorithm is based on a reinforcement learning scheme, namely Policy Iterations, and makes use of neural networks, in an Actor/Critic structure, to parametrically represent the control policy and the performance of the control system. The two neural networks are trained to express the optimal controller and optimal cost function which describes the infinite horizon control performance. Convergence of the algorithm is proven under the realistic assumption that the two neural networks do not provide perfect representations for the nonlinear control and cost functions. The result is a hybrid control structure which involves a continuous-time controller and a supervisory adaptation structure which operates based on data sampled from the plant and from the continuous-time performance dynamics. Such control structure is unlike any standard form of controllers previously seen in the literature. Simulation results, obtained considering two second-order nonlinear systems, are provided.

Inpainting and Zooming Using Sparse Representations

Abstract: Representing the image to be inpainted in an appropriate sparse representation dictionary, and combining elements from Bayesian statistics and modern harmonic analysis, we introduce an expectation maximization (EM) algorithm for image inpainting and interpolation. From a statistical point of view, the inpainting/interpolation can be viewed as an estimation problem with missing data. Toward this goal, we propose the idea of using the EM mechanism in a Bayesian framework, where a sparsity promoting prior penalty is imposed on the reconstructed coefficients. The EM framework gives a principled way to establish formally the idea that missing samples can be recovered/interpolated based on sparse representations. We first introduce an easy and efficient sparse-representation-based iterative algorithm for image inpainting. Additionally, we derive its theoretical convergence properties. Compared to its competitors, this algorithm allows a high degree of flexibility to recover different structural components in the image (piecewise smooth, curvilinear, texture, etc.). We also suggest some guidelines to automatically tune the regularization parameter.

Adaptive Reduced-Rank Processing Based on Joint and Iterative Interpolation, Decimation, and Filtering

Abstract: We present an adaptive reduced-rank signal processing technique for performing dimensionality reduction in general adaptive filtering problems. The proposed method is based on the concept of joint and iterative interpolation, decimation and filtering. We describe an iterative least squares (LS) procedure to jointly optimize the interpolation, decimation and filtering tasks for reduced-rank adaptive filtering. In order to design the decimation unit, we present the optimal decimation scheme and also propose low-complexity decimation structures. We then develop low-complexity least-mean squares (LMS) and recursive least squares (RLS) algorithms for the proposed scheme along with automatic rank and branch adaptation techniques. An analysis of the convergence properties and issues of the proposed algorithms is carried out and the key features of the optimization problem such as the existence of multiple solutions are discussed. We consider the application of the proposed algorithms to interference suppression in code-division multiple-access (CDMA) systems. Simulations results show that the proposed algorithms outperform the best known reduced-rank schemes with lower complexity.

Recovering Sparse Signals With a Certain Family of Nonconvex Penalties and DC Programming

Abstract: This paper considers the problem of recovering a sparse signal representation according to a signal dictionary. This problem could be formalized as a penalized least-squares problem in which sparsity is usually induced by a lscr1-norm penalty on the coefficients. Such an approach known as the Lasso or Basis Pursuit Denoising has been shown to perform reasonably well in some situations. However, it was also proved that nonconvex penalties like the pseudo lscrq-norm with q < 1 or smoothly clipped absolute deviation (SCAD) penalty are able to recover sparsity in a more efficient way than the Lasso. Several algorithms have been proposed for solving the resulting nonconvex least-squares problem. This paper proposes a generic algorithm to address such a sparsity recovery problem for some class of nonconvex penalties. Our main contribution is that the proposed methodology is based on an iterative algorithm which solves at each iteration a convex weighted Lasso problem. It relies on the family of nonconvex penalties which can be decomposed as a difference of convex functions (DC). This allows us to apply DC programming which is a generic and principled way for solving nonsmooth and nonconvex optimization problem. We also show that several algorithms in the literature dealing with nonconvex penalties are particular instances of our algorithm. Experimental results demonstrate the effectiveness of the proposed generic framework compared to existing algorithms, including iterative reweighted least-squares methods.

Delay-dependent stability and stabilization of neutral time-delay systems

Abstract: This paper is concerned with the problem of stability and stabilization of neutral time-delay systems. A new delay-dependent stability condition is derived in terms of linear matrix inequality by constructing a new Lyapunov functional and using some integral inequalities without introducing any free-weighting matrices. On the basis of the obtained stability condition, a stabilizing method is also proposed. Using an iterative algorithm, the state feedback controller can be obtained. Numerical examples illustrate that the proposed methods are effective and lead to less conservative results. Copyright © 2008 John Wiley & Sons, Ltd.

Convergence and Applications of Newton-type Iterations

Abstract: Recent results in local convergence and semi-local convergence analysis constitute a natural framework for the theoretical study of iterative methods. This monograph provides a comprehensive study of both basic theory and new results in the area. Each chapter contains new theoretical results and important applications in engineering, modeling dynamic economic systems, input-output systems, optimization problems, and nonlinear and linear differential equations. Several classes of operators are considered, including operators without Lipschitz continuous derivatives, operators with high order derivatives, and analytic operators. Each section is self-contained. Examples are used to illustrate the theory and exercises are included at the end of each chapter. The book assumes a basic background in linear algebra and numerical functional analysis. Graduate students and researchers will find this book useful. It may be used as a self-study reference or as a supplementary text for an advanced course in numerical functional analysis.

Consistent Depth Maps Recovery from a Video Sequence

Abstract: This paper presents a novel method for recovering consistent depth maps from a video sequence. We propose a bundle optimization framework to address the major difficulties in stereo reconstruction, such as dealing with image noise, occlusions, and outliers. Different from the typical multi-view stereo methods, our approach not only imposes the photo-consistency constraint, but also explicitly associates the geometric coherence with multiple frames in a statistical way. It thus can naturally maintain the temporal coherence of the recovered dense depth maps without over-smoothing. To make the inference tractable, we introduce an iterative optimization scheme by first initializing the disparity maps using a segmentation prior and then refining the disparities by means of bundle optimization. Instead of defining the visibility parameters, our method implicitly models the reconstruction noise as well as the probabilistic visibility. After bundle optimization, we introduce an efficient space-time fusion algorithm to further reduce the reconstruction noise. Our automatic depth recovery is evaluated using a variety of challenging video examples.

Matrices, Moments and Quadrature with Applications

Abstract: This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part.Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization.This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

Market-Based Generation and Transmission Planning With Uncertainties

Abstract: This paper presents a stochastic coordination of generation and transmission expansion planning model in a competitive electricity market. The Monte Carlo simulation method is applied to consider random outages of generating units and transmission lines as well as inaccuracies in the long-term load forecasting. The scenario reduction technique is introduced for reducing the computational burden of a large number of planning scenarios. The proposed model assumes a capacity payment mechanism and a joint energy and transmission market for investors' costs recovery. The proposed approach simulates the decision making behavior of individual market participants and the ISO. It is an iterative process for simulating the interactions among GENCOs, TRANSCOs and ISO. The iterative process might be terminated by the ISO based on a pre-specified stopping criterion. The case studies illustrate the applications of proposed stochastic method in a coordinated generation and transmission planning problem when considering uncertainties.

Finding the Largest Eigenvalue of a Nonnegative Tensor

Abstract: In this paper we propose an iterative method for calculating the largest eigenvalue of an irreducible nonnegative tensor. This method is an extension of a method of Collatz (1942) for calculating the spectral radius of an irreducible nonnegative matrix. Numerical results show that our proposed method is promising. We also apply the method to studying higher-order Markov chains.

Sensing-Based Spectrum Sharing in Cognitive Radio Networks

Abstract: In this paper, a new spectrum-sharing model called sensing-based spectrum sharing is proposed for cognitive radio networks. This model consists of two phases: In the first phase, the secondary user (SU) listens to the spectrum allocated to the primary user (PU) to detect the state of the PU; in the second phase, the SU adapts its transit power based on the sensing results. If the PU is inactive, the SU allocates the transmit power based on its own benefit. However, if the PU is active, the interference power constraint is imposed to protect the PU. Under this new model, the evaluation of the ergodic capacity of the SU is formulated as an optimization problem over the transmit power and the sensing time. Due to the complexity of this problem, two simplified versions, which are referred to as the perfect sensing case and the imperfect sensing case, are studied in this paper. For the perfect sensing case, the Lagrange dual decomposition is applied to derive the optimal power allocation policy to achieve the ergodic capacity. For the imperfect sensing case, an iterative algorithm is developed to obtain the optimal sensing time and the corresponding power allocation strategy. Finally, numerical results are presented to validate the proposed studies. It is shown that the SU can achieve a significant capacity gain under the proposed model, compared with that under the opportunistic spectrum access or the conventional spectrum sharing model.

MIMO Radar Waveform Optimization With Prior Information of the Extended Target and Clutter

Abstract: The concept of multiple-input multiple-output (MIMO) radar allows each transmitting antenna element to transmit an arbitrary waveform. This provides extra degrees of freedom compared to the traditional transmit beamforming approach. It has been shown in the recent literature that MIMO radar systems have many advantages. In this paper, we consider the joint optimization of waveforms and receiving filters in the MIMO radar for the case of extended target in clutter. A novel iterative algorithm is proposed to optimize the waveforms and receiving filters such that the detection performance can be maximized. The corresponding iterative algorithms are also developed for the case where only the statistics or the uncertainty set of the target impulse response is available. These algorithms guarantee that the SINR performance improves in each iteration step. Numerical results show that the proposed methods have better SINR performance than existing design methods.

Algorithms for entanglement renormalization

Abstract: We describe an iterative method to optimize the multiscale entanglement renormalization ansatz for the low-energy subspace of local Hamiltonians on a $D$-dimensional lattice. For translation-invariant systems the cost of this optimization is logarithmic in the linear system size. Specialized algorithms for the treatment of infinite systems are also described. Benchmark simulation results are presented for a variety of one-dimensional systems, namely, Ising, Potts, $XX$, and Heisenberg models. The potential to compute expected values of local observables, energy gaps, and correlators is investigated.

The MIMO Iterative Waterfilling Algorithm

Abstract: This paper considers the noncooperative maximization of mutual information in the vector Gaussian interference channel in a fully distributed fashion via game theory. This problem has been widely studied in a number of works during the past decade for frequency-selective channels, and recently for the more general multiple-input multiple-output (MIMO) case, for which the state-of-the art results are valid only for nonsingular square channel matrices. Surprisingly, these results do not hold true when the channel matrices are rectangular and/or rank deficient matrices. The goal of this paper is to provide a complete characterization of the MIMO game for arbitrary channel matrices, in terms of conditions guaranteeing both the uniqueness of the Nash equilibrium and the convergence of asynchronous distributed iterative waterfilling algorithms. Our analysis hinges on new technical intermediate results, such as a new expression for the MIMO waterfilling projection valid (also) for singular matrices, a mean-value theorem for complex matrix-valued functions, and a general contraction theorem for the multiuser MIMO watefilling mapping valid for arbitrary channel matrices. The quite surprising result is that uniqueness/convergence conditions in the case of tall (possibly singular) channel matrices are more restrictive than those required in the case of (full rank) fat channel matrices. We also propose a modified game and algorithm with milder conditions for the uniqueness of the equilibrium and convergence, and virtually the same performance (in terms of Nash equilibria) of the original game.

Feedforward control of piezoactuators in atomic force microscope systems

Abstract: This article describes an inversion-based feedforward approach to compensate for dynamic and hysteresis effects in piezoactuators with application to AFM technology. To handle the coupled behavior of dynamics and hysteresis, a cascade model is presented to enable the application of inversion-based feedforward control. The dynamics, which include vibration and creep, are modeled using linear transfer functions. A frequency-based method is used to invert the linear model to find an input that compensates for vibration and creep. The inverse is noncausal for nonminimum-phase systems. Similarly, the hysteresis is handled by an inverse-Preisach model. To avoid the complexity of finding the inverse-Preisach model, high- gain feedback control can be used to linearize the system's behavior. A feedforward input is then combined with the feedback system to compensate for the linear dynamics to achieve high-speed AFM imaging. Finally, recent efforts in feedforward control for an SPM application including the use of iteration to handle hysteresis as well as uncertainties and variations in the system model is discussed.

Robust ISAR Range Alignment via Minimizing the Entropy of the Average Range Profile

Abstract: In this letter, a novel global approach to range alignment for inverse synthetic aperture radar (ISAR) image formation is presented. The algorithm is based on the minimization of the entropy of the average range profile (ARP), and the processing chain is capable of exploiting the efficiency of the fast Fourier transform. With respect to the existing global methods, the new one requires no exhaustive search operation and eliminates the necessity of the parametric model for the relative offset among the range profiles. The derivation of the algorithm indicates that the presented methodology is essentially an iterative solution to a set of simultaneous equations, and its robustness is also ensured by the iterative structure. Some alternative criteria, such as the maximum contrast of the ARP, can be introduced into the algorithm with a minor change in the entropy-based method. The convergence and robustness of the presented algorithm have been validated by experimental ISAR data.