固定时间梯度流在l1-l2范数中的稀疏重构

压缩感知（compressed sensing,CS）是一种全新的信号采样技术,对于稀疏信号,它能够以远小于传统的Nyquist采样定理的采样点来重构信号。在压缩感知中, 采用动态连续系统,对l₁-l₂范数的稀疏信号重构问题进行了研究。提出了一种基于固定时间梯度流的稀疏信号重构算法,证明了该算法在Lyapunov意义上的稳定性并且收敛于问题的最优解。最后通过与现有的投影神经网络算法的对比,体现了该算法的可行性以及在收敛速度上的优势.     

基于CT图像重建的多重集合分裂可行性问题应用分析

为了较好地应用CQ算法解决稀疏角度CT 图像重建的问题,提出了一种新的实时的分块逐次混合算法．首先将稀疏角度CT 图像重建的重建问题转化成分裂可行性问题．其次,通过分析非空闭凸集C和Q的不同的定义,在N维实空间中分别针对不同的CQ算法给出了7种不同的实现方案．通过试验,分别对不同算法及其方案的重建精度和收敛速度进行了对比分析,并对多重集合分裂可行性问题算法中约束权因子的选取及其对输出的影响进行了研究,从而给出了CQ算法在稀疏角度CT图像重建问题中应用的最佳凸集定义方案．以此为基础,给出了所提出算法的最佳实现方案．试验结果表明,该算法收敛速度快,重建精度高,为多重集合分裂可行性问题及其改进算法在该重建问题上的应用提供了参考．     

Primal and dual alternating direction algorithms for ℓ 1-ℓ 1-norm minimization problems in compressive sensing

In this paper, we propose, analyze and test primal and dual versions of the alternating direction algorithm for the sparse signal reconstruction from its major noise contained observation data. The algorithm minimizes a convex non-smooth function consisting of the sum of ? ₁-norm regularization term and ? ₁-norm data fidelity term. We minimize the corresponding augmented Lagrangian function alternatively from either primal or dual forms. Both of the resulting subproblems admit explicit solutions either by using a one-dimensional shrinkage or by an efficient Euclidean projection. The algorithm is easily implementable and it requires only two matrix-vector multiplications per-iteration. The global convergence of the proposed algorithm is established under some technical conditions. The extensions to the non-negative signal recovery problem and the weighted regularization minimization problem are also discussed and tested. Numerical results illustrate that the proposed algorithm performs better than the state-of-the-art algorithm YALL1.     

不同凸集条件下CQ算法的应用分析

针对CQ算法,通过定义不同条件的下非空闭凸集C和Q,并结合讨论稀疏角度的CT重建问题,在R^N空间中给出了5种不同的实现方案,每种实现方案相对于CT重建模型,具备不同的物理含义.给定相同的迭代步数,通过仿真试验,分别对不同方案的重建精度进行了分析,从而确定了在相同收敛条件下CQ算法在应用时的最佳方案,为分裂可行性问题及其扩展形式在工程领域的应用提供了新的思路.     

A gradient projection method for the sparse signal reconstruction in compressive sensing

Many problems arising from machine learning, compressive sensing, linear inverse problem, and statistical inference involve finding sparse solutions to under-determined or ill-conditioned equations. In this paper, a gradient projection method is proposed to recover sparse signal in compressive sensing by solving the nonlinear convex constrained equations. The global convergence is established with the backtracking line search. Preliminary numerical experiments coping with the sparse signal reconstruction in compressive sensing are performed, which show that the proposed method is very effective and stable.     

Improved sampling and reconstruction in spline subspaces

As a special shift-invariant spaces, spline subspaces yield many advantages so that there are many practical applications for signal or image processing. In this paper, we pay attention to the sampling and reconstruction problem in spline subspaces. We improve lower bound of sampling set conditions in spline subspaces. Based on the improved explicit lower bound, a improved explicit convergence ratio of reconstruction algorithm is obtained. The improved convergence ratio occupies faster convergence rate than old one. At the end, some numerical examples are shown to validate our results.     

A GREEDY ALGORITHM FOR SPARSE PRECISION MATRIX APPROXIMATION

Precision matrix estimation is an important problem in statistical data analysis.This paper proposes a sparse precision matrix estimation approach,based on CLIME estimator and an efficient algorithm GISSρ that was originally proposed for l1 sparse signal recov-ery in compressed sensing.The asymptotic convergence rate for sparse precision matrix estimation is analyzed with respect to the new stopping criteria of the proposed GISSρ algorithm.Finally,numerical comparison of GISSρ with other sparse recovery algorithms,such as ADMM and HTP in three settings of precision matrix estimation is provided and the numerical results show the advantages of the proposed algorithm.     

求解低秩矩阵恢复问题的非单调交替梯度方向法

低秩矩阵恢复问题作为一类在图像处理和信号数据分析等领域中都十分重要的问题已被广泛研究.本文在交替方向算法的框架下,应用非单调技术,提出一种求解低秩矩阵恢复问题的新算法.该算法在每一步迭代过程中,首先利用一步带有变步长梯度算法同时更新低秩部分的两块变量,然后采用非单调技术更新稀疏部分的变量.在一定的假设条件下,本文证明了...     

修正HS共轭梯度法在大规模信号重构问题中的应用

本义研究了压缩感知在大规模信号恢复问题中应用的问题.利用修正HS共轭梯度法及光滑化方法,获得了具有较好重构效果的算法.数值实验表明用修正HS共轭梯度法解决大规模信号恢复问题是可行的.     

Relaxed two points projection method for solving the multiple-sets split equality problem

The multiple-sets split equality problem, a generalization and extension of the split feasibility problem, has a variety of specific applications in real world, such as medical care, image reconstruction, and signal processing. It can be a model for many inverse problems where constraints are imposed on the solutions in the domains of two linear operators as well as in the operators’ ranges simultaneously. Although, for the split equality problem, there exist many algorithms, there are but few algorithms for the multiple-sets split equality problem. Hence, in this paper, we present a relaxed two points projection method to solve the problem; under some suitable conditions, we show the weak convergence and give a remark for the strong convergence method in the Hilbert space. The interest of our algorithm is that we transfer the problem to an optimization problem, then, based on the model, we present a modified gradient projection algorithm by selecting two different initial points in different sets for the problem (we call the algorithm as two points algorithm). During the process of iteration, we employ subgradient projections, not use the orthogonal projection, which makes the method implementable. Numerical experiments manifest the algorithm is efficient.     

球面上$\ell_1$正则优化的随机临近梯度方法

本文研究球面上的$\ell_1$正则优化问题,其目标函数由一般光滑函数项和非光滑$\ell_1$正则项构成,且假设光滑函数的随机梯度可由随机一阶oracle估计.这类优化问题被广泛应用在机器学习,图像、信号处理和统计等领域.根据流形临近梯度法和随机梯度估计技术,提出一种球面随机临近梯度算法.基于非光滑函数的全局隐函数定理,分析了子问题解关于参数的Lipschtiz连续性,进而证明了算法的全局收敛性.在基于随机数据集和实际数据集的球面$\ell_1$正则二次规划问题、有限和SPCA问题和球面$\ell_1$正则逻辑回归问题上数值实验结果显示所提出的算法与流形临近梯度法、黎曼随机临近梯度法相比CPU时间上具有一定的优越性.     

A splitting primal-dual proximity algorithm for solving composite optimization problems

Our work considers the optimization of the sum of a non-smooth convex function and a finite family of composite convex functions, each one of which is composed of a convex function and a bounded linear operator. This type of problem is associated with many interesting challenges encountered in the image restoration and image reconstruction fields. We developed a splitting primal-dual proximity algorithm to solve this problem. Furthermore, we propose a preconditioned method, of which the iterative parameters are obtained without the need to know some particular operator norm in advance. Theoretical convergence theorems are presented. We then apply the proposed methods to solve a total variation regularization model, in which the L2 data error function is added to the L1 data error function. The main advantageous feature of this model is its capability to combine different loss functions. The numerical results obtained for computed tomography (CT) image reconstruction demonstrated the ability of the proposed algorithm to reconstruct an image with few and sparse projection views while maintaining the image quality.     

A SPARSE SUBSPACE TRUNCATED NEWTON METHOD FOR LARGE-SCALE BOUND CONSTRAINED NONLINEAR OPTIMIZATION

In this paper we report a sparse truncated Newton algorithm for handling large-scale simple bound nonlinear constrained minimixation problem. The truncated Newton method is used to update the variables with indices outside of the active set, while the projected gradient method is used to update the active variables. At each iterative level, the search direction consists of three parts, one of which is a subspace truncated Newton direction, the other two are subspace gradient and modified gradient directions. The subspace truncated Newton direction is obtained by solving a sparse system of linear equations. The global convergence and quadratic convergence rate of the algorithm are proved and some numerical tests are given.     

A variable fixing version of the two-block nonlinear constrained Gauss–Seidel algorithm for \ell _1-regularized least-squares

The problem of finding sparse solutions to underdetermined systems of linear equations is very common in many fields as e.g. signal/image processing and statistics. A standard tool for dealing with sparse recovery is the \(\ell _1\) -regularized least-squares approach that has recently attracted the attention of many researchers. In this paper, we describe a new version of the two-block nonlinear constrained Gauss–Seidel algorithm for solving \(\ell _1\) -regularized least-squares that at each step of the iteration process fixes some variables to zero according to a simple active-set strategy. We prove the global convergence of the new algorithm and we show its efficiency reporting the results of some preliminary numerical experiments.     

一种SENSE模型下信号重建的类-Dykstra近点有效算法

本文研究了SENSE模型下从部分傅里叶数据中信号的重建问题.利用类Dykstra近点方法和Bregman迭代方法,我们获得了一种SENSE模型下信号重建的加速类-Dykstra近点有效算法,并证明了该算法的收敛性.实验仿真显示,该方法比经典的分裂Bregman方法有效.     

Restricted Normal Cones and Sparsity Optimization with Affine Constraints

The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely behaved nonconvex relaxations. In this paper we consider the elementary method of alternating projections (MAP) for solving the sparsity optimization problem without employing convex heuristics. In a parallel paper we recently introduced the restricted normal cone which generalizes the classical Mordukhovich normal cone and reconciles some fundamental gaps in the theory of sufficient conditions for local linear convergence of the MAP algorithm. We use the restricted normal cone together with the notion of superregularity, which is inherently satisfied for the affine sparse optimization problem, to obtain local linear convergence results with estimates for the radius of convergence of the MAP algorithm applied to sparsity optimization with an affine constraint.     

Relaxed inertial proximal Peaceman-Rachford splitting method for separable convex programming

The strictly contractive Peaceman-Rachford splitting method is one of effective methods for solving separable convex optimization problem, and the inertial proximal Peaceman-Rachford splitting method is one of its important variants. It is known that the convergence of the inertial proximal Peaceman-Rachford splitting method can be ensured if the relaxation factor in Lagrangian multiplier updates is underdetermined, which means that the steps for the Lagrangian multiplier updates are shrunk conservatively. Although small steps play an important role in ensuring convergence, they should be strongly avoided in practice. In this article, we propose a relaxed inertial proximal Peaceman-Rachford splitting method, which has a larger feasible set for the relaxation factor. Thus, our method provides the possibility to admit larger steps in the Lagrangian multiplier updates. We establish the global convergence of the proposed algorithm under the same conditions as the inertial proximal Peaceman-Rachford splitting method. Numerical experimental results on a sparse signal recovery problem in compressive sensing and a total variation based image denoising problem demonstrate the effectiveness of our method.     

On asymptotic convergence of the block-iterative Fisher scoring algorithm

This short paper studies convergence properties, particularly asymptotic convergence, of the block-iterative Fisher scoring (BFS) algorithms recently proposed by Ma and Hudson (2008). While applicable in other inverse problem domains (e.g. astronomy, geophysics, signal processing or remote sensing), this class of algorithms was designed for tomographic image reconstruction from projections in medicine. A BFS algorithm is used to reconstruct the patient’s internal structural or functional activity from collected projection data. We briefly introduce the BFS algorithm and a general convergence result provided in Ma and Hudson (2008). This result is used to prove the asymptotic convergence of two specific BFS algorithms under new conditions.     

An Augmented ADMM Algorithm With Application to the Generalized Lasso Problem

In this article, we present a fast and stable algorithm for solving a class of optimization problems that arise in many statistical estimation procedures, such as sparse fused lasso over a graph, convex clustering, and trend filtering, among others. We propose a so-called augmented alternating direction methods of multipliers (ADMM) algorithm to solve this class of problems. Compared to a standard ADMM algorithm, our proposal significantly reduces the computational cost at each iteration while maintaining roughly the same overall convergence speed. We also consider a new varying penalty scheme for the ADMM algorithm, which could further accelerate the convergence, especially when solving a sequence of problems with tuning parameters of different scales. Extensive numerical experiments on the sparse fused lasso problem show that the proposed algorithm is more efficient than the standard ADMM and two other existing state-of-the-art specialized algorithms. Finally, we discuss a possible extension and some interesting connections to two well-known algorithms. Supplementary materials for the article are available online.     

QP-Free,Truncated Hybrid Methods for Large-Scale Nonlinear Constrained Optimization

1.IntroductionInthispaperweconsiderthefollowingnonlinearprogrammingproblemminimizef(x)subjecttogj(x)2o,jEJ={1,...,m}.(1'1)Extensionstoproblemincludingalsoequalityconstraintswillbepossible.Thefunctionf:W-Rlandgj:Rn-R',jEJaretwicecontinuouslydifferentiable.Inpaxticular,weapplyQP-free(withoutquadraticprogrammingsubproblems),truncatedhybridmethodsforsolvingthelarge-scaJenonlinearprogrammingproblems,inwhichthenumberofvariablesandthenumberofconstraiotsin(1.1)aregreat.Wediscussthecase,wheresecon…