压缩感知：交替优化投影矩阵设计新算法

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"该文提出了一种基于交替优化的压缩感知投影矩阵设计高效算法，旨在为特定类别的信号，这些信号可以通过预设的字典Ψ稀疏表示，优化设计投影矩阵Φ。目标是使等效字典ΦΨ的Gram矩阵与一组放松的等角紧框架(ETFs)之间的距离最小化。并给出了一种针对给定Gram矩阵的最优投影矩阵设计的有效方法。" 在压缩感知(Compressive Sensing, CS)领域，有效的投影矩阵设计对于信号的高效恢复至关重要。这篇研究论文探讨了如何通过交替优化(Alternating Optimization)策略来实现这一目标。交替优化是一种优化技术，它通过反复迭代两个或多个变量，使得整体目标函数逐步收敛到最优解。在这个问题中，变量包括投影矩阵Φ和信号的字典Ψ。文章的核心是寻找能够最佳重构信号的投影矩阵。投影矩阵Φ的设计是为了确保信号在被压缩采样后，仍能有效地进行重构。通常，这需要矩阵Φ与一个理想的字典（如ETFs）具有良好的兼容性。ETFs是一类特殊构造的框架，其元素之间有恒定且较小的夹角，这有利于信号的稀疏表示和恢复。论文提出了一个新的优化目标，即最小化等效字典ΦΨ的Gram矩阵与一组放松的等角紧框架之间的距离。Gram矩阵反映了矩阵乘积的内积性质，可以度量两个字典之间的相似度。通过优化这个距离，可以确保投影矩阵与信号的稀疏表示字典之间有更好的匹配，从而提高信号恢复的准确性和效率。为了实现这个优化目标，作者们开发了一种有效的方法来设计最优的投影矩阵，特别是当给定一个固定的Gram矩阵时。这种方法可能涉及到迭代更新投影矩阵的元素，直到达到某种收敛条件，比如目标函数的改变小于某个阈值或者达到预定的迭代次数。这篇论文对压缩感知中的投影矩阵设计问题进行了深入研究，提出了一种新的优化算法，并提供了具体的实现策略。这一工作对于提高压缩感知在图像处理、通信和数据采集等领域的性能有着重要的理论和实际意义。

novel algorithm can be found in Section 3. An extension of

designing projection matrix to the case with sparse

representation error is shown in Section 4. Detailed dis-

cussions about how this cost function is developed and

how this problem is efﬁciently solved can be also found in

Section 4.InSection 5, simulations are given to illustrate

the effectiveness and efﬁciency of the methods shown in

this paper. The conclusions are given in Section 6 to end

this paper.

2. Preliminaries and problem formulation

Following [20],wedeﬁne the maximal mutual coherence

of the matrix D as

ðÞ

9 max

1 r i a j r L

jDð: ; iÞ

Dð: ; jÞj

J Dð: ; iÞJ

J Dð: ; jÞJ

: ð6Þ

Denote

9diagðGð1; 1Þ

 1=2

⋯Gðk; kÞ

 1=2

⋯GðL; LÞ

 1=2

Then, the Gram matrix of

D ¼ DS

,denotedasG,isnor-

malized such that

Gðk; kÞ¼1; 8 k. Obviously, μ

ðDÞ¼

max

i a j

jGði; jÞj. The lower and upper bounds of μ

ðDÞ

are [23]

μ r μ

ðDÞr 1

with

μ 9

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

L  M

MðL  1Þ

. The av erag e mutual coherence used in

[20] is given by

DðÞ9

8ði;jÞ A H

jGði; jÞj

ð7Þ

where N

is the number of entries in the index set

9fði; jÞ: jGði; jÞjZ ζ&iajg with 0r ζ o 1thatwecan

choose as desired. Both μ

and μ

are used as the measures

of projection matrix design in this paper.

Frames are an important concept in signal analysis. We

refer the reader to [29,30] for the detailed discussions on

frames. Here, we only introduce two important classes of

frames: tight frames and ETFs, which are necessary for our

later analysis. The sequence of column vectors Dð: ; jÞ of

matrix D is a frame for the Hilbert space R

if there exist

two constants 0o α r β o þ1 such that

αJ v J

r J D

r β J v J

ð8Þ

for all v A R

N1

. Such a frame is called α-tight if α ¼ β in (8).

A normalized frame fDð: ; jÞg ði:e:; J Dð: ; jÞJ

¼ 1; 8 jÞ is

said to be equiangular if

jDð: ; iÞ

Dð: ; jÞj ¼ c; 8iaj

where c is a positive constant. As shown in [23], a matrix D

with J Dð: ; jÞJ

¼ 1; 8 j achieves μ

ðDÞ¼μ if and only if

fDð: ; jÞg is ETF, moreover, μ

ðDÞ¼μ can only hold if

Lr Mð M þ 1Þ=2. It is clear that the orthogonal bases form a

special class of ETFs. An ETF has a very nice average mutual

coherence behavior and has been used in optimal dic-

tionary design [24]. In addition, such an idea was also

extended to the optimal projection matrix design [25,26].

The projection matrix design problem based on the ETF

structure can be formulated as [25,26]

min

;Φ

J G

Ψ

ΦΨ J

A H

¼fG

¼ G

; diagðG

Þ¼1; max

i a j

ði; jÞj r ζgð9Þ

where ζ is a parameter to control the maximal non-

diagonal value in G

, diag ðG

Þ denotes the diagonal entries

of G

and 1 represents a vector of proper dimension with

its elements all equal to 1. It can be found that (9) is a

highly nonconvex problem. The alternating method has

been widely used to address such problems [23,24,26]. The

basic idea is to update Φ and G

alternatively. Firstly, the

projection matrix Φ is ﬁxed and G

is updated. Secondly, G

is ﬁxed and the projection matrix Φ is updated. This pro-

ceeding will be repeated until a stop criterion is reached.

At the ﬁrst stage, we directly project the Gram matrix of

ΦΨ to the set H

. The key problem is the second step.

Abolghasemi et al. [26] proposed a gradient descent

method to address the problem in this step. For con-

venience, the procedures of solving (9) in [26] is denoted

as Algorithm 1 and shown as follows.

Algorithm 1. AFS.

Initialization:

Sparsify dictionary matrix Ψ, initialize Φ

to a random matrix, and

the number of iterations: Iter

outer

Output:

Projection matrix Φ.

Iteration:

1: Φ’Φ

2: for l ¼1 to Iter

outer

3: ðG’Ψ

ΦΨÞ

4: Update G

5: diagðG

Þ’1

6: for all iaj do

7: G

ði; jÞ’

Gði; jÞ if jGði; jÞjr ζ

ζ  signðGði; jÞÞ otherwise

(

where sign ðÞ is a sign function.

8: end for

9: Update Φ:

10: Φ’

Φ ¼ arg min

J G

Ψ

ΦΨ J

ð10Þ

11: end for

Note that the gradient descent algorithm can only

reach a local minimum for (10). Moreover, as pointed out

by the authors of [26], the accuracy of the solution for (10)

will affect the ﬁnal performance in terms of signal recov-

ery accuracy. Hence, in order to reach a good solution, a

large number of iterations is needed for the gradient

descent algorithm while the step size must be chosen

carefully. Meanwhile, the complexity of the algorithm is

increased. According to these observations, a new algo-

rithm is proposed in this paper which can yield a better

solution for (10) with a much lower complexity when the

dimension of the problem is not very large. In order to

carry out the proposed algorithm, some manipulations

should be taken to cast (10) to a processable equation. The

related manipulations are shown as follows.

In the case of G

¼ I, this problem is equivalent to (5).

T. Hong et al. / Signal Processing 125 (2016) 9–20 11

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