Bregman迭代的统一 primal-dual 算法框架在信号处理中的应用
需积分: 9 69 浏览量
更新于2024-07-24
收藏 808KB PDF 举报
"这篇文章提出了一种统一的原对偶算法框架，基于Bregman迭代，应用于信号和图像处理中的两类问题。它与现有的方法建立了联系，特别是与线性化Bregman、Split Bregman等Bregman迭代方法有关。论文证明了通用算法框架在温和假设下的收敛性，并通过ℓ1基追踪、TV-L2最小化和矩阵填充的应用进行了演示。"
这篇由Xiaoqun Zhang、Martin Burger和Stanley Osher合著的论文详细介绍了压缩感知领域的Bregman迭代方法。Bregman迭代是一种解决优化问题的技术，特别适用于处理具有稀疏解的信号和图像恢复问题，如压缩感知。压缩感知是一种理论，允许从较少的测量中重构高维信号，前提是信号在某种基或框架下是稀疏的。
文中提出的统一原对偶算法框架，旨在解决两类由信号和图像处理应用产生的问题。原对偶方法是优化问题求解中的一个强大工具，通过同时更新原问题和对偶问题的变量，可以在保证解的质量的同时提高计算效率。这种框架不仅统一了不同的Bregman迭代方法，还扩展了它们的适用范围。
线性化Bregman是Bregman迭代的一种变体，它通过引入线性化步骤来加速收敛速度，特别是在处理L1范数正则化的优化问题时，如L1基追踪（也称为稀疏表示）。L1基追踪问题在信号去噪和压缩感知中有广泛应用，因为它能诱导出稀疏解。
TV-L2最小化，即Total Variation (TV)与L2范数的组合，常用于图像去噪和重建。TV范数鼓励图像的平滑部分保持连续，而L2范数则处理图像的局部变化，这样的结合可以有效地保留图像边缘同时去除噪声。
Split Bregman方法则是另一种有效的优化策略，它将复杂的优化问题分解为更简单的子问题，从而简化了求解过程。这种方法在处理包含L1和L2范数项的优化问题时表现出色。
论文的核心贡献在于证明了这个通用算法框架在适度假设下的收敛性，这意味着算法能够在一定条件下稳定地找到问题的解。此外，通过实际应用示例——矩阵填充，展示了算法的有效性。矩阵填充问题涉及从不完全观测的数据中恢复完整矩阵，这对于推荐系统和数据恢复等应用至关重要。
"Primal and Dual Bregman Method"提供了一个强大的工具箱，用于处理各种信号和图像处理中的优化挑战，通过统一的原对偶算法框架，将不同类型的Bregman迭代方法集成在一起，增强了算法的灵活性和实用性。
J Sci Comput (2011) 46: 20–46 25
where q(y) =inf
x∈X
L(x;y). The pair (x; y) is a saddle point of the Lagrangian function
L(x; y) if and only if
x is an optimal solution of the primal problem (2.2)andy is a
solution of the dual problem (2.3) respectively. In general, the existence of solutions for
a primal problem does not imply the existence of saddle points of Lagrangian functions,
but the converse holds, see [30]. Here, we assume the existence of solutions to the original
primal problems and related saddle point problems.
Classical methods based on the gradient/subgradient for solving the above saddle point
problems have been of great interest since the seminal work of Arrow, Hurwicz and
Uzawa [1]. These methods alternate a minimization of the Lagrangian function L(x; y)
with respect to x given y and gradient ascent with respect to y given x.Moreprecisely,a
general Uzawa method can be written as
x
k+1
=argmin
x∈X
L(x, y
k
),
y
k+1
=P
Y
(y
k
+γ∂
y
L(x
k+1
; y
k
))
(2.4)
where ∂
y
L(x
k+1
; y
k
) is a subgradient of L(x; y) with respect to y at (x
k+1
; y
k
), P
Y
(·)
is the orthogonal projection operator to Y and γ is a positive number. The convergence
is established assuming strict convexity. This method and its variants are largely studied
for quadratic programming in the literature due to the simplicity. This method is in fact
a gradient-type algorithm and under certain conditions, a linear convergence rate can be
achieved [30]. Particularly, a class of inexact Uzawa methods are proposed and analyzed
in [6] for linear saddle points problems by introducing preconditioning matrices in order to
achieve fast convergence. Our proposed methods are also based on the idea of the general
Uzawa method and we attempt to achieve the convergence without strict convexity assump-
tions.
Now, we consider two general convex programming problems which cover various signal
and image processing applications:
(P
0
) min
x∈R
N
J
0
(x) s.t. Ax =b (2.5)
as in (1.3)and
(P
1
) min
x∈R
N
,z∈R
K
J
1
(z) +H(x)
s.t. Bx =z (2.6)
as (1.14), where J
0
:R
N
→(−∞, ∞], J
1
:R
K
→(−∞, ∞] and H :R
N
→(−∞, ∞] are
closed proper convex functionals, and A ∈R
M×N
, B ∈R
K×N
.
The Lagrangian function for problem (2.5) and problem (2.6) are respectively
L
0
(x; y) =J
0
(x) −y,Ax −b (2.7)
and
L
1
(x, z; y) =J
1
(z) +H(x)−y,z −Bx, (2.8)
where y is the Lagrange multiplier associated with the respective constraint in each prob-
lem. Note that the multiplier variables in the two problems are in different sets R
M
and
R
K
, but here we use the same notation y without confusion. The related dual problems are
respectively defined as
(D
0
) min
y
J
∗
0
(A
y) −b, y, (2.9)
剩余26页未读，继续阅读
点击了解资源详情
点击了解资源详情
点击了解资源详情
2021-02-11 上传
2021-05-29 上传
2019-12-28 上传
2020-05-14 上传
2019-01-30 上传
2021-02-07 上传
u011118681
- 粉丝: 0
- 资源: 3
我的内容管理
展开
最新资源
- MATLAB新功能：Multi-frame ViewRGB制作彩色图阴影
- XKCD Substitutions 3-crx插件：创新的网页文字替换工具
- Python实现8位等离子效果开源项目plasma.py解读
- 维护商店移动应用：基于PhoneGap的移动API应用
- Laravel-Admin的Redis Manager扩展使用教程
- Jekyll代理主题使用指南及文件结构解析
- cPanel中PHP多版本插件的安装与配置指南
- 深入探讨React和Typescript在Alias kopio游戏中的应用
- node.js OSC服务器实现：Gibber消息转换技术解析
- 体验最新升级版的mdbootstrap pro 6.1.0组件库
- 超市盘点过机系统实现与delphi应用
- Boogle: 探索 Python 编程的 Boggle 仿制品
- C++实现的Physics2D简易2D物理模拟
- 傅里叶级数在分数阶微分积分计算中的应用与实现
- Windows Phone与PhoneGap应用隔离存储文件访问方法
- iso8601-interval-recurrence：掌握ISO8601日期范围与重复间隔检查
