利用分片贪婪算法求解分片稀疏恢复问题

在实际应用中,有一些信号是具有分片的结构的.本文我们提出一种分片正交匹配追踪算法(P\_OMP)来求解分片稀疏恢复问题,旨在保护分片信号中的分片结构(或者小尺度非零元).P\_OMP算法是基于CoSaMP和OMMP算法的思想上延伸出的一种针对分片稀疏问题的贪婪算法. P\_OMP算法不仅仅具有OMP算法的优势,还能够在比CoSaMP方法更松弛的条件下得到同样的误差下降速率.进一步,P\_OMP~算法在保护分片稀疏信号的尺度细节信息上表现的更好.数值实验表明相比于CoSaMP, OMP, OMMP和BP算法, P\_OMP算法在分片稀疏恢复上更有效更稳定.     

压缩感知和稀疏优化简介

介绍压缩感知和稀疏优化的基本概念、理论基础和算法概要. 压缩感知利用原始信号的稀疏性,从远少于信号元素个数的测量出发,通过求解稀疏优化问题来恢复完整的原始稀疏信号. 通过一个小例子展示这一过程,并以此说明压缩感知和稀疏优化的基本理念. 接着简要介绍用以保证l₁凸优化恢复稀疏信号的零空间性质和RIP条件. 最后介绍求解稀疏优化的几个经典算法.     

异步并行矩阵多分裂块松弛迭代算法

1 引言 众所周知,许多微分方程经过差分或有限元离散,即可归结为线性代数方程组 Ax=b,A∈L(R~n)非奇异,x,b∈R~n.(1.1)缘于原问题的物理特性,系数矩阵A∈L(R~n)通常是大型稀疏的,并且具有规则的分块结构。鉴此,文[1]基于矩阵多重分裂的概念,并运用线性迭代法的松弛加速技巧,提出了求解这类大型稀疏分块线性代数方程组的并行矩阵多分裂块松弛迭代算法,并在适当的条件下建立了算法的收敛理论。对于SIMD多处理机系统,这类算法是颇为适用和行之有效的。     

无约束的l2,1-分析法重构冗余紧框架下分块稀疏信号的条件

文章主要利用分块稀疏信号的凸分解技术分析无约束的l_2,1-分析模型,建立无约束的l_2,1-分析法重构冗余紧框架下分块稀疏信号的条件,其条件基于紧框架下的限制等距性质.首先,利用分块稀疏信号的凸分解技术建立两个重要技术引理.其次,基于发展的两个技术引理建立无约束的l_2,1-分析法恢复冗余紧框架下分块稀疏信号新的恢复条件,其条件基于紧框架下的限制等距性质,改进了现存最好的恢复条件.最后,设计数值实验,说明无约束的l_2,1-分析法重构冗余紧框架下分块稀疏信号的性能.     

三类有效预处理子的关系及其优化

针对一类由时谐抛物方程约束的最优控制问题导出的分块$2\times2$复线性方程组,进一步研究了三类有效的块预处理子,推导了这三类预处理子间的关系,结论表明三个预处理矩阵的特征值由同一个矩阵确定.通过分析预处理矩阵的谱性质,获得了有效的参数选择策略,可以进一步改进和优化现有结果,同时获得了预处理矩阵的精确特征值分布,并证明了此结果是目前文献中最优结果.最后,给出实例,不仅验证了优化的预处理子和迭代方法的有效性,而且说明了理论结果是令人信服的.     

弱化限制等距性的稀疏信号恢复

众所周知,传统的信号压缩和重建遵循香农一耐奎斯特采样定律,即采样率必须至少为信号最高频率的两倍,才能保证在重建时不产生失真,这无疑将给信号采样,传输和存储过程带来越来越大的压力.随着科技的飞速发展,特别是近年来传感器技术获取数据能力提高,物联网等促使人类社会的数据规模遽增,大数据时代正式到来.大数据的规模效应给数据存储,传输,管理以及数据分析带来了极大的挑战.压缩采样应运而生.限制等距性(Restricted Isometry Property,RIP)在压缩传感中起着关键的作用.只有满足限制等距条件的压缩矩阵才能平稳恢复原始信号.RIP作为衡量矩阵是否能作为测量矩阵得到了认可,但是此理论的缺陷在于对任一矩阵,很难有通用,快速的算法来验证其是否满足RIP条件.很多学者尝试弱化RIP条件以找到测量矩阵构造的突破口.首先构造了新的限制等距条件δ_(1.5k)+θ_(k,1.5k)≤1,然后证明在这个条件下无噪声稀疏信号能被精确的恢复,并且噪声稀疏信号能被平稳的估计.最后,通过比较表明δ_(1.5k)+θ_(k,1.5k)≤1优于现存的条件.     

基于迭代投影的梯度硬阈值追踪算法

梯度硬阈值追踪算法是求解稀疏优化问题的有效算法之一.考虑到算法中投影对最优解的影响,提出一种比贪婪策略更好的投影算法是很有必要的.针对一般的稀疏约束优化问题,利用整数规划提出一种迭代投影策略,将梯度投影算法中的投影作为一个子问题求解.通过迭代求解该子问题得到投影的指标集,并以此继续求解原问题,以提高梯度硬阈值追踪算法的计算效果.证明了算法的收敛性,并通过数值实例验证了算法的有效性.     

半监督距离度量学习内蕴加速投影梯度算法

考虑求解一类半监督距离度量学习问题. 由于样本集(数据库)的规模与复杂性的激增, 在考虑距离度量学习问题时, 必须考虑学习来的距离度量矩阵具有稀疏性的特点. 因此, 在现有的距离度量学习模型中, 增加了学习矩阵的稀疏约束. 为了便于模型求解, 稀疏约束应用了Frobenius 范数约束. 进一步, 通过罚函数方法将Frobenius范数约束罚到目标函数, 使得具有稀疏约束的模型转化成无约束优化问题. 为了求解问题, 提出了正定矩阵群上加速投影梯度算法, 克服了矩阵群上不能直接进行线性组合的困难, 并分析了算法的收敛性. 最后通过UCI数据库的分类问题的例子, 进行了数值实验, 数值实验的结果说明了学习矩阵的稀疏性以及加速投影梯度算法的有效性.     

不等式约束优化一个可行序列线性方程组算法

提出了求解非线性不等式约束优化问题的一个可行序列线性方程组算法. 在每次迭代中, 可行下降方向通过求解两个线性方程组产生, 系数矩阵具有较好的稀疏性. 在较为温和的条件下, 算法具有全局收敛性和强收敛性, 数值试验表明算法是有效的.     

两阶段特殊结构混合0-1规划的分解算法

本文介绍了一种用于求解具有特殊结构的两阶段混合0-1规划问题的原始-对偶分解算法,并以CPLEX软件作为核心求解器将算法实现.该算法将原问题分解成两个相对简单的子问题,较传统分解算法有更平衡的分解结构和收敛性.实验数据表明,该算法在求解较大规模、稀疏度较大、耦合度较大的复杂两阶段下三角结构混合0-1规划问题时,相比CPLEX提供的分枝剪枝法,在时间效率上有明显提高.算法最后通过固定0-1变量的取值可以得到满足管理精度要求的近似最优解.     

Structured backward error for palindromic polynomial eigenvalue problems

A detailed structured backward error analysis for four kinds of palindromic polynomial eigenvalue problems (PPEP)

On the Diophantine equation : Higher-order recurrences

Let be a field of characteristic and let be a linear recurring sequence of degree in defined by the initial terms and by the difference equation

with . Finally, let be an element of . In this paper we are giving fairly general conditions depending only on on , and on under which the Diophantine equation

has only finitely many solutions . Moreover, we are giving an upper bound for the number of solutions, which depends only on . This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.

     

On the Number of Topological Types Occurring in a Parameterized Family of Arrangements

Let be an o-minimal structure over ℝ, a closed definable set, and

Schur–Weyl Theory for C*‐algebras

To each irreducible infinite dimensional representation $(\pi ,\mathcal {H})$ of a C*‐algebra $\mathcal {A}$, we associate a collection of irreducible norm‐continuous unitary representations $\pi _{\lambda }^\mathcal {A}$ of its unitary group ${\rm U}(\mathcal {A})$, whose equivalence classes are parameterized by highest weights in the same way as the irreducible bounded unitary representations of the group ${\rm U}_\infty (\mathcal {H}) = {\rm U}(\mathcal {H}) \cap (\mathbf {1} + K(\mathcal {H}))$ are. These are precisely the representations arising in the decomposition of the tensor products $\mathcal {H}^{\otimes n} \otimes (\mathcal {H}^*)^{\otimes m}$ under ${\rm U}(\mathcal {A})$. We show that these representations can be realized by sections of holomorphic line bundles over homogeneous Kähler manifolds on which ${\rm U}(\mathcal {A})$ acts transitively and that the corresponding norm‐closed momentum sets $I_{\pi _\lambda ^\mathcal {A}}^{\bf n} \subseteq {\mathfrak u}(\mathcal {A})^{\prime }$ distinguish inequivalent representations of this type.     

The generalized Roper-Suffridge extension operator in Banach spaces (III)

Suppose that X is a complex Banach space with the norm ‖·‖ and n is a positive integer with dim X ⩾ n ⩾ 2. In this paper, we consider the generalized Roper-Suffridge extension operator $ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f) $ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f) on the domain $ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } $ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } defined by

Bicomplex Quantum Mechanics: I. The Generalized Schrödinger Equation

We introduce the set of bicomplex numbers which is a commutative ring with zero divisors defined by where We present the conjugates and the moduli associated with the bicomplex numbers. Then we study the bicomplex Schr?dinger equation and found the continuity equations. The discrete symmetries of the system of equations describing the bicomplex Schr?dinger equation are obtained. Finally, we study the bicomplex Born formulas under the discrete symmetries. We obtain the standard Born’s formula for the class of bicomplex wave functions having a null hyperbolic angle.     

由H\"{o}rmander向量场构成的抛物方程的 $W_{\ast}^{1,p}$ 正则性

设 $X_{1},\cdots ,X_{q}\ (q相似文献   

ON ADMISSffilLITY OF VARIANCE COMPONENTS ESTIMATES

Suppose that there is a variance components model $$\[\left\{ {\begin{array}{*{20}{c}} {E\mathop Y\limits_{n \times 1} = \mathop X\limits_{n \times p} \mathop \beta \limits_{p \times 1} }\{DY = \sigma _2^2{V_1} + \sigma _2^2{V_2}} \end{array}} \right.\]$$ where $\[\beta \]$,$\[\sigma _1^2\]$ and $\[\sigma _2^2\]$ are all unknown, $\[X,V > 0\]$ and $\[{V_2} > 0\]$ are all known, $\[r(X) < n\]$. The author estimates simultaneously $\[(\sigma _1^2,\sigma _2^2)\]$. Estimators are restricted to the class $\[D = \{ d({A_1}{A_2}) = ({Y^''}{A_1}Y,{Y^''}{A_2}Y),{A_1} \ge 0,{A_2} \ge 0\} \]$. Suppose that the loss function is $\[L(d({A_1},{A_2}),(\sigma _1^2,\sigma _2^2)) = \frac{1}{{\sigma _1^4}}({Y^''}{A_1}Y - \sigma _1^2) + \frac{1}{{\sigma _2^4}}{({Y^''}{A_2}Y - \sigma _2^2)^2}\]$. This paper gives a necessary and sufficient condition for $\[d({A_1},{A_2})\]$ to be an equivariant D-asmissible estimator under the restriction $\[{V_1} = {V_2}\]$, and a sufficient condition and a necessary condition for $\[d({A_1},{A_2})\]$ to equivariant D-asmissible without the restriction.     

Required Number of Iterations for Sparse Signal Recovery via Orthogonal Least Squares

In countless applications, we need to reconstruct a $K$-sparse signal $\mathbf{x}\in\mathbb{R}^n$ from noisy measurements $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{v}$, where $\mathbf{\Phi}\in\mathbb{R}^{m\times n}$ is a sensing matrix and $\mathbf{v}\in\mathbb{R}^m$ is a noise vector. Orthogonal least squares (OLS), which selects at each step the column that results in the most significant decrease in the residual power, is one of the most popular sparse recovery algorithms. In this paper, we investigate the number of iterations required for recovering $\mathbf{x}$ with the OLS algorithm. We show that OLS provides a stable reconstruction of all $K$-sparse signals $\mathbf{x}$ in $\lceil2.8K\rceil$ iterations provided that $\mathbf{\Phi}$ satisfies the restricted isometry property (RIP). Our result provides a better recovery bound and fewer number of required iterations than those proposed by Foucart in 2013.     

树的导出匹配覆盖

The induced matching cover number of a graph G without isolated vertices,denoted by imc(G),is the minimum integer k such that G has k induced matchings M1,M2,…,Mk such that,M1∪M2 ∪…∪Mk covers V(G).This paper shows if G is a nontrivial tree,then imc(G) ∈ {△*0(G),△*0(G) + 1,△*0(G)+2},where △*0(G) = max{d0(u) + d0(v) :u,v ∈ V(G),uv ∈ E(G)}.