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在通信系统中，尤其是无线通信领域，数据传输经常会受到信道衰落的影响，导致信号质量下降。为了提高传输效率和可靠性，人们发展了一系列均衡技术，其中Turbo均衡是一种结合了迭代译码和均衡的高效方法。本文将深入探讨基于GAMP（Generalized Approximate Message Passing）算法的Turbo均衡在信道未知信号未知条件下的应用，以及如何在MATLAB环境中实现这一技术。 GAMP算法是一种用于处理大规模稀疏系统的统计推断方法，它源自消息传递算法（Message Passing Algorithm）家族，特别是由Belief Propagation算法发展而来。在GAMP中，主要处理的是高维稀疏信号的恢复问题，特别适用于大规模线性系统。在通信系统中，GAMP可以用于估计信道状态信息（Channel State Information, CSI）和接收信号，从而实现信道均衡。 Turbo均衡是一种融合了前向纠错编码（Forward Error Correction, FEC）和均衡器的迭代处理方式。它利用信道编码的冗余信息来改善均衡效果，同时均衡器的输出也会反馈到解码器，增强解码性能。在信道未知且信号未知的情况下，Turbo均衡需要通过迭代的方式逐步学习和校正信道的影响，以达到最佳的均衡效果。在MATLAB环境中实现alternating_GAMP算法，首先需要理解GAMP的基本步骤，包括初始化、消息传递和更新规则。这些步骤通常涉及矩阵操作、概率密度函数的计算以及梯度下降等优化方法。MATLAB提供了强大的数学工具箱，如Signal Processing Toolbox和Optimization Toolbox，为实现GAMP算法提供了便利。在实现Turbo均衡时，需要构建以下关键模块： 1. 信道模型：根据实际通信环境设置，如多径衰落、频率选择性衰落等。 2. GAMP模块：执行GAMP算法，估计信道系数和接收信号。 3. 解码器模块：采用turbo码或其他FEC码进行解码，与GAMP模块交互进行迭代。 4. 均衡器模块：根据GAMP提供的信道估计进行均衡处理，改善符号检测的性能。 5. 循环控制：设定迭代次数，判断收敛条件，决定何时停止迭代。在MATLAB代码实现中，可以使用结构化编程，将每个模块封装成独立的函数，然后通过主程序调用并协调它们之间的交互。同时，为了提高计算效率，可以考虑利用MATLAB的并行计算工具箱进行并行化处理。 alternating_GAMP算法是针对信道未知且信号未知情况的一种有效解决方案，通过结合GAMP的强大信号恢复能力和Turbo均衡的迭代优化，能够在MATLAB环境下实现高效的通信系统均衡。正确理解和实现这一算法对于提升通信系统的性能至关重要。

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共 4082 条

SPAMS: a SPArse Modeling Software, v2.3

Julien Mairal

julien.mairal@m4x.org

June 14, 2012

Contents

1 Introduction 2

2 Installation 4

3 Dictionary Learning and Matrix Factorization Toolbox 4

3.1 Function mexTrainDL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2 Function mexTrainDL_Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.3 Function nmf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.4 Function nnsc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Sparse Decomposition Toolbox 12

4.1 Function mexOMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 Function mexOMPMask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3 Function mexLasso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4 Function mexLassoWeighted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.5 Function mexLassoMask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.6 Function mexCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.7 Function mexSOMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.8 Function mexL1L2BCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.9 Function mexSparseProject . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Proximal Toolbox 27

5.1 Regularization Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Function mexProximalFlat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.3 Function mexProximalTree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.4 Function mexProximalGraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.5 Function mexProximalPathCoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.6 Function mexEvalPathCoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.7 Problems Addressed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.7.1 Regression Problems with the Square Loss . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.7.2 Classiﬁcation Problems with the Logistic Loss . . . . . . . . . . . . . . . . . . . . . . . . 41

5.7.3 Multi-class Classiﬁcation Problems with the Softmax Loss . . . . . . . . . . . . . . . . . 41

5.7.4 Multi-task Regression Problems with the Square Loss . . . . . . . . . . . . . . . . . . . 41

5.7.5 Multi-task Classiﬁcation Problems with the Logistic Loss . . . . . . . . . . . . . . . . . 42

5.7.6 Multi-task and Multi-class Classiﬁcation Problems with the Softmax Loss . . . . . . . . 42

5.8 Function mexFistaFlat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.9 Function mexFistaTree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.10 Function mexFistaGraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.11 Function mexFistaPathCoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6 Miscellaneous Functions 58

6.1 Function mexConjGrad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.2 Function mexBayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.3 Function mexCalcAAt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.4 Function mexCalcXAt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.5 Function mexCalcXY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.6 Function mexCalcXYt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.7 Function mexCalcXtY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.8 Function mexInvSym . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.9 Function mexNormalize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.10 Function mexSort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.11 Function mexDisplayPatches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.12 Function mexCountPathsDAG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.13 Function mexRemoveCyclesGraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.14 Function mexCountConnexComponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

A Duality Gaps with Fenchel Duality 66

A.0.1 Duality Gaps without Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

A.0.2 Duality Gaps with Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

1 Introduction

SPAMS (SPArse Modeling Software) is an open-source optimization toolbox under licence GPLv3. It im-

plements algorithms for solving various machine learning and signal processing problems involving sparse

regularizations.

The library is coded in C++, is compatible with Linux, Mac, and Windows 32bits and 64bits Operating

Systems. It is interfaced with Matlab, but can be called from any C++ application. A R and Python interface

has been developed by Jean-Paul Chieze.

It requires an implementation of BLAS and LAPACK for performing eﬃcient linear algebra operations

such as the one provided by matlab/R, atlas, netlib, or the one provided by Intel (Math Kernel Library). It

also exploits multi-core CPUs when this feature is supported by the compiler, through OpenMP.

The current licence is GPLv3 available at http://www.gnu.org/licenses/gpl.html, which limits its

usage. For other usages (such as the use in proprietary softwares), please contact the author.

Version 2.3 of SPAMS is divided into three “toolboxes” and has a few additional miscellaneous functions:

• The Dictionary learning and matrix factorization toolbox contains the online learning technique

of [19, 20] and its variants for solving various matrix factorization problems:

– dictionary Learning for sparse coding;

– sparse principal component analysis (seen as a sparse matrix factorization problem);

– non-negative matrix factorization;

– non-negative sparse coding.

• The Sparse decomposition toolbox contains eﬃcient implementations of

– Orthogonal Matching Pursuit, (or Forward Selection) [32, 24];

– the LARS/homotopy algorithm [27, 8] (variants for solving Lasso and Elastic-Net problems);

– a weighted version of LARS;

– OMP and LARS when data come with a binary mask;

– a coordinate-descent algorithm for `

-decomposition problems [11, 9, 33];

– a greedy solver for simultaneous signal approximation as deﬁned in [31, 30] (SOMP);

– a solver for simulatneous signal approximation with `

-regularization based on block-coordinate

descent;

– a homotopy method for the Fused-Lasso Signal Approximation as deﬁned in [9] with the homotopy

method presented in the appendix of [20];

– a tool for projecting eﬃciently onto a few convex sets inducing sparsity such as the `

-ball using the

method of [3, 17, 7], and Elastic-Net or Fused Lasso constraint sets as proposed in the appendix

of [20].

• The Proximal toolbox: An implementation of proximal methods (ISTA and FISTA [1]) for solving

a large class of sparse approximation problems with diﬀerent combinations of loss and regularizations.

One of the main features of this toolbox is to provide a robust stopping criterion based on duality gaps

to control the quality of the optimization, whenever possible. It also handles sparse feature matrices for

large-scale problems. The following regularizations are implemented:

– Tikhonov regularization (squared `

-norm);

– `

-norm, `

, `

∞

-norms;

– Elastic-Net [35];

– Fused Lasso [29];

– tree-structured sum of `

-norms (see [14, 15]);

– tree-structured sum of `

∞

-norms (see [14, 15]);

– general sum of `

∞

-norms (see [21, 22]);

– mixed `

-norms on matrices [34, 26];

– mixed `

∞

-norms on matrices [34, 26];

– mixed `

-norms on matrices plus `

[28, 10];

– mixed `

∞

-norms on matrices plus `

;

– group-lasso with `

or `

∞

-norms;

– group-lasso+`

;

– multi-task tree-structured sum of `

∞

-norms (see [21, 22]);

– trace norm;

– `

pseudo-norm (only with ISTA);

– tree-structured `

(only with ISTA);

– rank regularization for matrices (only with ISTA);

– the path-coding penalties of [23].

All of these regularization functions can be used with the following losses

– square loss;

– square loss with missing observations;

– logistic loss, weighted logistic loss;

– multi-class logistic.

This toolbox can also enforce non-negativity constraints, handle intercepts and sparse matrices. There

are also a few additional undocumented functionalities, which are available in the source code.

• A few tools for performing linear algebra operations such as a conjugate gradient algorithm, manipulating

sparse matrices and graphs.

The toolbox was written by Julien Mairal when he was at INRIA, with the collaboration of Francis Bach

(INRIA), Jean Ponce (Ecole Normale Supérieure), Guillermo Sapiro (University of Minnesota), Guillaume

Obozinski (INRIA) and Rodolphe Jenatton (INRIA).

R and Python interfaces have been written by Jean-Paul Chieze (INRIA), and a few contributors have

helped us making compilation scripts for various platforms.

2 Installation

The toolbox used to come with pre-compiled binaries for various plateforms. Now the sources are available

and you will have to compile it yourself.

The user has the choice of the BLAS library, but the Intel MKL is recommended for the best performance.

Note that the builtin blas library of Matlab is a version of the Intel MKL (not the most recent one though),

and oﬀers a good performance.

The folder doc contains the documentation in pdf and html. build/ contains the binary ﬁles, including

the help for each command. test_release contains various matlab scripts which call the diﬀerent functions

of this toolbox.

The software package comes also with a script bash that has to be used to launch matlab for Linux and/or

Mac OS versions. The install procedure is described in a ﬁle called HOW_TO_INSTALL and in the ﬁle compile.m

for the Matlab version.

3 Dictionary Learning and Matrix Factorization Toolbox

This is the section for dictionary learning and matrix factorization, corresponding to [19, 20].

3.1 Function mexTrainDL

This is the main function of the toolbox, implementing the learning algorithms of [20]. Given a training set

, . . . ,. It aims at solving

min

D∈C

lim

n→+∞

i=1

min



− Dα

+ ψ(α

)



. (1)

ψ is a sparsity-inducing regularizer and C is a constraint set for the dictionary. As shown in [20] and in the help

ﬁle below, various combinations can be used for ψ and C for solving diﬀerent matrix factorization problems.

What is more, positivity constraints can be added to α as well. The function admits several modes for choosing

the optimization parameters, using the parameter-free strategy proposed in [19], or using the parameters t

and ρ presented in [20]. Note that for problems of a reasonable size, and when ψ is the `

-norm,

the function mexTrainDL_Memory can be faster but uses more memory.

% Usage : [D [ model ]] = mexTrainDL (X, param [ , model ] ) ;

% model i s o p t i o n a l

% Name: mexTrainDL

% D e s c r i p t i o n : mexTrainDL i s an e f f i c i e n t implemen tatio n o f the

% d i c t i o n a r y l e a r n i n g tec hni q ue p r ese n ted i n

% " Online Lea rning f o r Matrix F a c t o r i z a t i o n and S parse Coding "

% by J u l i e n Mairal , F ran c is Bach , Jean Ponce and Guillermo Sap iro

% arXiv : 0 9 08.0050

% " Online D icti o n ary Le arnin g f o r Sparse Coding "

% by J u l i e n Mairal , F ran c is Bach , Jean Ponce and Guillermo Sap iro

% ICML 200 9.

% Note that i f you use param . mode=1 or 2 , i f the t r a i n i n g s e t has a

% r e a s o n a b le s i z e and you have enough memory on your computer , you

% sh ould use mexTra inDL_Memory i n s t e a d .

% I t a d d r e s s e s the d i c t i o n a r y l e a r n i n g problems

% 1) i f param . m ode=0

% min_{D i n C} (1/n) sum_{ i =1}^n ( 1/2 ) | | x_i−Dalpha_i | | _2^2 s . t . . . .

% | | alpha_i | | _1 <= lambda

% 2) i f param . m ode=1

% min_{D i n C} (1/n) sum_{ i =1}^n | | alpha_i | | _1 s . t . . . .

% | | x_i−Dalpha_i | | _2^2 <= lambda

% 3) i f param . m ode=2

% min_{D i n C} (1/n) sum_{ i =1}^n ( 1/2 ) | | x_i−Dalpha_i | | _2^2 + . . .

% 4) i f param . m ode=3, the sparse coding i s done with OMP

% min_{D i n C} (1/n) sum_{ i =1}^n ( 1/2 ) | | x_i−Dalpha_i | | _2^2 s . t . . . .

% | | alpha_i | | _0 <= lambda

% 5) i f param . m ode=4, the sparse coding i s done with OMP

% min_{D i n C} (1/n) sum_{ i =1}^n | | alpha_i | | _0 s . t . . . .

% | | x_i−Dalpha_i | | _2^2 <= lambda

% 6) i f param . m ode=5, the sparse coding i s done with OMP

% min_{D i n C} (1/n) sum_{ i =1}^n 0 . 5 | | x_i−Dalpha_i | | _2^2 +lambda | | alpha_i | | _0

%% C i s a convex s e t v e r i f y i n g

% 1) i f param . modeD=0

% C={ D in Real^{m x p} s . t . f o r a l l j , | | d_j | | _2^2 <= 1 }

% 2) i f param . modeD=1

% C={ D in Real^{m x p} s . t . f o r a l l j , | | d_j | | _2^2 + . . .

% gamma1 | | d_j | | _1 <= 1 }

% 3) i f param . modeD=2

% C={ D in Real^{m x p} s . t . f o r a l l j , | | d_j | | _2^2 + . . .

% gamma1 | | d_j | | _1 + gamma2 FL( d_j ) <= 1 }

% 4) i f param . modeD=3

% C={ D in Real^{m x p} s . t . f o r a l l j , (1−gamma1) | | d_j | | _2 ^2 + . . .

% gamma1 | | d_j | | _1 <= 1 }

% P o t e n t i a l l y , n can be very l a r g e with t h i s alg ori thm .

% I nputs : X: double m x n matrix ( inp ut s i g n a l s )

% m i s t he s i g n a l s i z e

% n i s t he number o f s i g n a l s t o decompose

% param : s t r u c t

% param .D: ( o p t i o n a l ) doub le m x p matrix ( d i c t i o n a r y )

% p i s t he number o f eleme nt s i n the d i c t i o n a r y

% When D i s not provided , the d i c t i o n a r y i s i n i t i a l i z e d

% with random e le ments from th e t r a i n i n g s e t .

% param .K ( s i z e o f th e d i c t i o n a r y , o p t i o n a l i s param .D i s provided )

% param . lambda ( parameter )

% param . lambda2 ( o pti o n a l , by d e f a u l t 0)

% param . i t e r ( number of i t e r a t i o n s ) . I f a n e g ative number i s

% p ro vi de d i t w i l l perform th e computation d ur in g the

% corr e spon d ing number of s eco nd s . For i n s t a n c e param . i t e r =−5

% l e a r n s the d i c t i o n a r y during 5 seco nd s .

% param . mode ( o p t iona l , s e e above , by d e f a u l t 2)

% param . posAlpha ( o p t ion a l , adds p o s i t i v i t y c o n s t r a i n t s on the

% c o e f f i c i e n t s , f a l s e by default , not compatible with

% param . mode =3 ,4)

% param . modeD ( opti o n al , s e e above , by d e f a u l t 0)

% param . posD ( o p tion a l , adds p o s i t i v i t y c o n s t r a i n t s on the

% d i c t i o nary , f a l s e by def a u l t , not co mpatible with

% param . modeD= 2)

% param . gamma1 ( o p t i o n a l parameter f o r param . modeD >= 1)

% param . gamma2 ( o p t i o n a l parameter f o r param . modeD = 2)

% param . b a t c h s i z e ( o ptio n a l , s i z e of th e minibatch , by d e f a u l t

% 512 )

% param . iter_updateD ( opti o n al , number o f BCD i t e r a t i o n s f o r the d i c t i o n a r y

% update step , by d e f a u l t 1)

% param . modeParam ( o p t i m i z ation mode) .

% 1) i f param . modeParam=0, th e o p t i m i z a t ion us e s the

% parameter f r e e s t r a t e g y o f the ICML paper

% 2) i f param . modeParam=1, th e o p t i m i z a t ion us e s the

% parameters rho as i n arXiv : 0 908.0050

% 3) i f param . modeParam=2, th e o p t i m i z a t ion us e s e x p o nential

% decay wei gh ts with up dates of th e form

% A_{ t } <− rho A_{ t −1} + alpha_t alpha_t ^T

% param . rho ( o p t i o n a l ) tuning parameter ( s e e paper arXiv : 0 9 0 8 . 0 0 5 0 )

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bbyouyou

2024-07-22

资源有很好的参考价值，总算找到了自己需要的资源啦。
LyrenLeigh

2022-05-03

用户下载后在一定时间内未进行评价，系统默认好评。
qydsj

2021-11-15

用户下载后在一定时间内未进行评价，系统默认好评。
邓哈哈哈哈

2022-12-25

资源很不错，内容和描述一致，值得借鉴，赶紧学起来！
NetLover123

2022-08-29

这个资源总结的也太全面了吧，内容详实，对我帮助很大。

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