正则化可理解为一种罚函数法摘要
稀疏和低秩特性是大多数信号所具有的潜在低维结构模式，它们为数据表达与分析、揭示事物内在本质属性和知识理解提供了契机。稀疏是指信号自身的非零元个数或信号在某个变换域内的非零项表示系数的个数远小于其维度。作为稀疏概念的推广，低秩则是指矩阵的秩（非零奇异值的个数）远小于矩阵的维度。从可能含噪的低维测量中获得稀疏解的过程称为稀疏重构，而使用少量且非冗余的低秩因子矩阵去捕获（可能含有缺失信息的）高维数据矩阵的主成分则称为低秩逼近。稀疏重构和低秩逼近已广泛地应用于信号处理、无线通信、模式识别、机器学习及计算机视觉等领域，高效、鲁棒及可扩展性的稀疏重构与低秩逼近算法是实现其应用的前提条件。
近年来，压缩感知（Compressed Sensing）理论的兴起与发展促使许多学者开始致力于稀疏重构与低秩逼近算法的研究。虽然相关算法研究已取得了一系列重要成果，但现有大多数算法具有一定的限制或假设条件，针对某些实际应用场景，这些算法并不适用。本论文在稀疏重构和低秩逼近算法方面开展进一步研究，主要研究内容及创新点概括如下：
（1）针对非均匀噪声下的单字典多测量矢量（Multiple Measurement Vectors, MMV）联合稀疏支集重构问题，本论文提出了一种新颖的单字典协方差拟合算法。首先，从MMV测量模型的二阶统计协方差矩阵出发，通过矢量化运算和线性变换操作，得到不含未知噪声功率的单测量矢量（Single Measurement Vector,
SMV）稀疏表示模型。其次，利用样本协方差矩阵误差的渐近高斯分布特性进行预白化处理，得到了标准高斯噪声下的SMV模型，并通过求解非负稀疏优化问题重构稀疏支集。最后，通过分析模型的平方误差分布特性，讨论了正则化参数选择问题。针对不同的测量矩阵，还讨论了算法的可辨识性问题，并从理论上证明所提算法能够实现稀疏度大于测量维数（欠定）的稀疏支集重构。随机测量矩阵和窄带波达方向（Direction-of-Arrival,DOA）估计中的实验结果表明该算法在低信噪比和较大差异的非均匀噪声下具有比现有算法更好的重构性能。
（2）针对非均匀噪声下的多字典MMV联合稀疏支集重构问题，本论文提出了一种新颖的多字典协方差拟合算法。首先，从不同测量字典下的多个MMV模型的协方差矩阵出发，通过矢量化操作，得到新颖的多字典SMV稀疏表示模型。然后，通过对每一个字典下的SMV模型作线性变换和预白化处理，消除了多字典模型的未知非均匀噪声功率对稀疏重构的影响。最后，通过加权协方差拟合标准
和ℓ2,1混合范数稀疏正则化准则，构建了一个稳健的非负联合稀疏重构凸优化问题，克服了字典中的原子相关性问题。同时，利用对偶优化理论，给出了概率意义下的正则化参数选择。随机测量矩阵和宽带DOA估计中的实验结果表明，该算法不仅能实现欠定稀疏支集重构，还具有解模糊作用。
（3）针对脉冲噪声下的联合稀疏信号与字典参数重构的线谱估计问题，本论文通过使用平滑的ℓp(0<p<2)范数对脉冲噪声进行拟合，并结合基于对数求和的正则化稀疏罚函数，将字典基不匹配的线谱
估计问题转化为一个鲁棒性最优化问题，该问题具有非凸性且含有稀疏变量和（非线性）字典参数变量。为了求解此非凸优化问题，我们提出了一种基于优化最小化（Majorization Minimization, MM）的迭代重加权算法，该算法的每一次迭代解都为原目标函数在前一次迭代点处的二次上界逼近函数的近似解。仿真实验结果表明，所提算法在脉冲噪声下具有比现有算法更优越的性能。
（4）针对脉冲噪声下的低秩逼近问题，本论文结合平滑的ℓp(0<p<2)范数误差准则和矩阵分解思想，将该问题描述为一个具有可变正则化的非凸优化问题。基于块优化最小化（Block MM）方法，我们提出了一种可扩展且比较灵活的块迭代重加权算法框架。该算法不仅能适用于对缺失样本的重建，还可以直接得到具有结构化的（稀疏或半稀疏）非负矩阵分解，从而能够充分挖掘矩阵的内部结构信息并扩展其实际应用场景。本论文还从理论上证明了所提算法产生的迭代序列具有全局收敛性且能收敛到原问题的一个稳定点。通过在矩阵填充（Matrix Completion）、非负特征提取和运动恢复结构（Structure From Motion）等应用的仿真实验结果表明，与现有同类算法相比，该算法在脉冲噪声和高样本缺损率情况下具有较大的性能提升。
关键词：稀疏重构，压缩感知，低秩逼近，矩阵分解，正则化
ABSTRACT
Sparsity and low-rank are the potential low-dimensional structure patterns of most natural signals.Thes
e characteristics provide an opportunity for data representation and analysis or knowledge understanding to unveil the inherent structure of the world.A sig-nal(vector)is called sparse if the number of nonzero elements in itself or the nonzero representation coefficients in a transform domain is much smaller than its dimension.As the extension of the sparsity,low-rank refers to the fact that the rank of a matrix(the number of nonzero singular values)is less than the the dimension(the number of rows or columns)of the associated matrix.The process of obtaining sparse solutions from (possibly noisy)low-dimensional measurements of a signal is called the sparse recon-struction,whereas using a small number of nonredundant low-rank factors to capture the principal components of the high dimensional data matrix(which may contain missing information)is referred to as the low-rank approximation.Sparse reconstruction and low-rank approximation have been widely used in many practical applications,such as signal processing,wireless communication,pattern recognition,machine learning,computer vision,etc.Pursuing efficient,robust and scalable sparse reconstruction and low-rank approximation algorithms are the foundations of their applications.
In recent years,the development of compressed sensing theory has led many schol-ars to study sparse reconstruction and low-rank approximation algorithms.Although a multitude of relevant algorithms have been reported,the existing approaches rest upon some key restrictions or assumption
s and may fall short or even break down in some practical scenarios.This dissertation concentrates on the algorithm aspect of sparse re-construction and low-rank approximation in nonuniform noise or impulsive noise envi-ronment,and the main contributions are summarized as follows.
(1)For the problem of sparse support reconstruction from multiple measurement vectors(MMV)with a single dictionary,a novel sparse covariancefitting algorithm is proposed for the nonuniorm noise environment.Firstly,from the second-order covari-ance matrix of the MMV model,a single measurement vector(SMV)model without any knowledge of the unknown noise power is obtained by exploiting vectorization and linear transformation.Secondly,the asymptotic Gaussian distribution of the sample covariance matrix error is exploited to perform a prewhitening processing,and hence an SMV model
is available under the standard Gaussian noise,and then the problem of the sparse support reconstruction is formulated as a nonnegative sparse optimization problem.Finally,the square error distribution of the SMV model is analyzed,and the regularization parameter selection is also discussed in a probability sense.The identifiability of the proposed algo-rithm is also discussed for different measurement matrices,and it is theoretically proved that the proposed algorithm can achieve the sparse support recovery with a larger sparse support than the measurement dimension.The simulation results in both the random mea-surement matrix and narrowband direction-of-arrival(DOA)estimation ill
ustrate that the proposed algorithm has better reconstruction performance than existing approaches in nonuniform noise environment.
(2)For the problem of sparse support reconstruction from MMV with multiple sin-gle dictionary,a novel sparse covariancefitting algorithm is developed for the nonuniorm noise environment.Firstly,from the second-order covariance matrices of multiple MMV models with multiple dictionaries,many SMV representation models are obtained by vectorizing the convariance matrices.Then,the unknown nonuniform noise power are eliminated through the linear transformation and prewhitening of these SMV models.Fi-nally,a robust weighted sparse optimization problem is formulated by using the mixed ℓ2,1-norm,which can overcome the ambiguity issue resulting from the dictionary corre-lation.Based on the duality optimization theory,the regularization parameter selection is also discussed.The simulation results in both the random measurement matrix and wideband DOA estimation demonstrate that the proposed algorithm can not only achieve the sparse support reconstruction with larger sparse support,but can also address the ambiguity problem.
(3)For the line spectral estimation problem of joint sparse signal and parameter re-construction under impulsive noise,a robust formulation is proposed,which includes the generalizedℓp-norm(0<p<2)data-fidelityfitting term added to a log-sum sparsity-promoting regularizer.The proposed formulation involves,
besides the unknown sparse signal,a highly nonlinear frequency parameter.To handle this intractable nonconvex problem,an iteratively reweightedℓ2approach is developed via majorizing the original objective function by a quadratic surrogate function.Each iteration is an approximate solution of the quadratic upper bound surrogate of the original objective function at the previous iteration.Simulation results illustrate that the proposed approach attains a sig-nificant performance improvement over the existing methods under impulsive noise.
(4)For the problem of low-rank matrix approximation in the presence of impulsive noise and missing data,a unified formulation is proposed from the point of view of matrix factorization,which involves minimizing anℓp-norm(0<p<2)based data-fidelity term plus structure-promoting constraints,including nonnegativity and/or sparsity.To han-dle this computationally intractable problem,a block iteratively reweighted algorithmic framework is developed based on the block majorization minimization.Each iteration of our algorithm framework is obtained by minimizing a regularized reweightedℓ2surro-gate function with a closed-form minimizer in an interweaved block update fashion.The proposed approach can directly be utilized for(sparse or semi-sparse)nonnegative ma-trix factorization.Meanwhile,the nontrivial global convergence property of the proposed algorithm is also proved:the whole iteration sequence is convergent and converges to a stationary point.Numerical experiments using both synthetic data and real data demon-strate the superiority of the proposed algorithm.
Keywords:Sparse reconstruction,compressed sensing,low-rank approximation,matrix factorization,regularization

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