l1-magic用户指南：高效解决稀疏信号恢复的MATLAB实现
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更新于2024-07-28
收藏 277KB PDF 举报 "l1-magic工具箱用户指南是一份详细的文档，旨在介绍如何通过凸优化技术在高度不完整的数据中恢复稀疏信号。这份指南由Emmanuel Candès和Justin Romberg撰写，来自加州理工学院，发布于2005年10月。该工具箱与一系列论文相辅相成，这些论文发展了从少量线性测量中重构稀疏向量的理论，即使在存在噪声的情况下也能实现，如b = Ax0 + e。
l1-magic提供了一个MATLAB代码库，处理七种不同的场景，证明了这种信号恢复方法在大规模问题上的计算可行性，即使数据点数量达到数百万。主要问题可以分为两类：一类是可以通过线性规划（LPs）形式化的问题，另一类是通过二次序锥编程（SOCPs）处理的问题。对于LPs，采用了通用的路径跟踪的 primal-dual 方法来求解，而对于SOCPs，则采用了一种通用的对数_barrier算法，其实现遵循了文献[2]中的第11章。
该工具箱的目标不仅是展示最新技术，而是作为一个概念验证，展示了即使在大规模问题中，这些信号恢复程序也能够有效、快速地执行。通过使用l1-magic，用户可以探索稀疏信号的高效重构，这对于信号处理、压缩感知、机器学习等领域具有重要意义，因为稀疏性是许多现代数据集的重要特性，如图像处理中的自然图像、语音信号分析以及基因表达数据等。在使用过程中，用户需要理解如何正确设置参数，选择适当的模型，以及如何解读和应用恢复得到的稀疏解，以便在实际应用中获得最佳效果。"
In a nutshell, the primal dual algorithm finds the optimal z
?
(along with optimal dual vectors
ν
?
and λ
?
) by solving this system of nonlinear equations. The solution procedure is the classical
Newton method: at an interior point (z
k
, ν
k
, λ
k
) (by which we mean f
i
(z
k
) < 0, λ
k
> 0), the
system is linearized and solved. However, the step to new point (z
k+1
, ν
k+1
, λ
k+1
) must be
modified so that we remain in the interior.
In practice, we relax the complementary slackness condition λ
i
f
i
= 0 to
λ
k
i
f
i
(z
k
) = −1/τ
k
, (2)
where we judiciously increase the parameter τ
k
as we progress through the Newton iterations.
This biases the solution of the linearized equations towards the interior, allowing a smooth, well-
defined “central path” from an interior point to the solution on the boundary (see [15,20] for an
extended discussion).
The primal, dual, and central residuals quantify how close a point (z, ν, λ) is to satisfying (KKT )
with (2) in place of the slackness condition:
r
dual
= c
0
+ A
T
0
ν +
X
i
λ
i
c
i
r
cent
= −Λf − (1/τ )1
r
pri
= A
0
z − b,
where Λ is a diagonal matrix with (Λ)
ii
= λ
i
, and f =
f
1
(z) . . . f
m
(z)
T
.
From a point (z, ν, λ), we want to find a step (∆z , ∆ν, ∆λ) such that
r
τ
(z + ∆z, ν + ∆ν, λ + ∆λ) = 0. (3)
Linearizing (3) with the Taylor expansion around (z, ν, λ),
r
τ
(z + ∆z, ν + ∆ν, λ + ∆λ) ≈ r
τ
(z, ν, λ) + J
r
τ
(z, νλ)
∆z
∆ν
∆λ
,
where J
r
τ
(z, νλ) is the Jacobian of r
τ
, we have the system
0 A
T
0
C
T
−ΛC 0 −F
A
0
0 0
∆z
∆v
∆λ
= −
c
0
+ A
T
0
ν +
P
i
λ
i
c
i
−Λf − (1/τ)1
A
0
z − b
,
where m × N matrix C has the c
T
i
as rows, and F is diagonal with (F )
ii
= f
i
(z). We can
eliminate ∆λ using:
∆λ = −ΛF
−1
C∆z − λ − (1/τ)f
−1
(4)
leaving us with the core system
−C
T
F
−1
ΛC A
T
0
A
0
0
∆z
∆ν
=
−c
0
+ (1/τ)C
T
f
−1
− A
T
0
ν
b − A
0
z
. (5)
With the (∆z, ∆ν, ∆λ) we have a step direction. To choose the step length 0 < s ≤ 1, we ask
that it satisfy two criteria:
1. z + s∆z and λ + s∆λ are in the interior, i.e. f
i
(z + s∆z) < 0, λ
i
> 0 for all i.
2. The norm of the residuals has decreased sufficiently:
kr
τ
(z + s∆z, ν + s∆ν, λ + s∆λ)k
2
≤ (1 − αs) · kr
τ
(z, ν, λ)k
2
,
where α is a user-s precified parameter (in all of our implementations, we have set α = 0.01).
4
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