题目：压缩感知重构算法之迭代硬阈值(Iterative Hard Thresholding,IHT)

本篇来介绍IHT重构算法。一般在压缩感知参考文献中，提到IHT时一般引用的都是文献【1】，但IHT实际上是在文献【2】中提出的。IHT并不是一种凸优化算法，它类似于OMP，是一种迭代算法，但它是由一个优化问题推导得到的。文献【1】和文献【2】的作者相同，署名单位为英国爱丁堡大学(University ofEdinburgh)，第一作者的个人主页见参考文献【3】，从个人主页来看，作者现在已到英国南安普敦大学(University of Southampton)，作者发表的论文均可以从其个人主页中下载。

文献【1】的贡献是当把IHT应用于压缩感知重构问题时进行了一个理论分析：

1、迭代硬阈值(IHT)的提出

值得一提的是，IHT在文献【2】中提出时并不叫Iterative Hard Thresholding，而是M-Sparse Algorithm，如下图所示：

该算法是为了求解M-稀疏问题(M-sparse problem)式(3.1)而提出的，经过一番推导得到了迭代公式式(3.2)，其中H_M(·)的含义参见式(3.3)。

这里面最关键的问题是：式(3.2)这个迭代公式是如何推导得到的呢？

以下Step1~Step4推导过程可以参见本文的补充说明：迭代硬阈值(IHT)的补充说明，若要透彻地理解IHT，需要知道Majorization-Minimization优化框架和硬阈值(Hard Thresholding)函数。

2、Step1:替代目标函数

首先，将式(3.1)的目标函数用替代目标函数(surrogate objective fucntion)式(3.5)替换：

这里中的M应该指的是M-sparse，S应该指的是Surrogate。这里要求：

为什么式目标函数式(3.1)可以用式(3.5) 替代呢？这得往回看一下了……

实际上，文献【2】分别针对两个优化问题进行了讨论，本篇主要是文献中的第二个优化问题，由于两个问题有一定的相似性，所以文中在推导第二个问题时进行了一些简化，下面简单回顾一些必要的有关第一个问题的内容，第一个优化问题是：

将目标函数定义为：

为了推导迭代公式（详见式(2.2)和式(2.3)）式(1.5)用如下替代目标函数代替：

这里注意波浪下划线中提到的“[29]”(参见文献【4】)，surrogate objective function的思想来自这篇文件。然后注意对Φ的约束（第一个红框），之后以会有这个约束，个人认为是为了使式(2.5)后半部分大于等于零，即为了使

大于等于零（当y=z时这部分等于零）。由此自然就有了式(2.5)与式(1.5)两个目标函数的关系（第二个红框），这也很容易理解，将y=z代入式(2.5)自然可得这个关系。

到此应该明白式(2.5)为什么可以替代式(1.5)了吧……

而我们用式(3.5)替代目标函数

的道理是一模一样的。

补充一点：有关对||Φ||2<1的约束文献【2】中有一处提到了如下描述：

3、Step2:替代目标函数变形

接下来，式(3.5)进行了变形：

这个式子是怎么来的呢？我们对式(3.5)进行一下推导：

这里，后面三项2范数的平方是与y无关的项，因此可视为常量，若对参数y求最优化时这三项并不影响优化结果，可略去，因此就有了变形的结果，符号“∝”表示成正比例。

4、Step3:极值点的获得

接下来文献【2】直接给出了极值点：

注意文中提到了“landweder”，搜索一下可知经常出现的是“landweder迭代”，这个暂且不提。那么极值点是如何推导得到的呢？其实就是一个简单的配方，中学生就会的：

令，则

当，取得最小值

5、Step4:迭代公式的获得

极值点得到了，替代目标函数的极小值也得到了：

那么如何得到迭代公式式(3.2)呢？这时要注意，推导过程中有一个约束条件一直没管，即式(3.1)中的约束条件：

也就是向量y的稀疏度不大于M。综合起来说，替代函数的最小值是

那么怎么使这个最小值在向量y的稀疏度不大于M的约束下最小呢，显然是保留最大的M项（因为是平方，所以要取绝对值absolute value），剩余的置零（注意这里有个负号，所以要保留最大的M项）。

至此，我们就得到了迭代公式式(3.2)。

6、IHT算法的MATLAB代码

这里一共给出三个版本的IHT实现：

第一个版本：

在作者的主页有官方版IHT算法MATLAB代码，但有些复杂，这里给出一个简化版的IHT代码，方便理解：

function [ y ] = IHT_Basic( x,Phi,M,mu,epsilon,loopmax )
%IHT_Basic Summary of this function goes here
%Version: 1.0 written by jbb0523 @2016-07-30
%Reference:Blumensath T, Davies M E. Iterative Thresholding for Sparse Approximations[J].
%Journal of Fourier Analysis & Applications, 2008, 14(5):629-654.
%(Available at: http://link.springer.com/article/10.1007%2Fs00041-008-9035-z)
%   Detailed explanation goes here  if nargin < 6  loopmax = 3000;  end  if nargin < 5    epsilon = 1e-3;    end   if nargin < 4    mu = 1;    end   [x_rows,x_columns] = size(x);    if x_rows<x_columns    x = x';%x should be a column vector    end  n = size(Phi,2);  y = zeros(n,1);%Initialize y=0  loop = 0;  while(norm(x-Phi*y)>epsilon && loop < loopmax)  y = y + Phi'*(x-Phi*y)*mu;%update y  %the following two lines of code realize functionality of H_M(.)  %1st: permute absolute value of y in descending order  [ysorted inds] = sort(abs(y), 'descend');  %2nd: set all but M largest coordinates to zeros  y(inds(M+1:n)) = 0;  loop = loop + 1;  end
end

第二个版本：（作者给出的官方版本）

文件：hard_l0_Mterm.m(\sparsify_0_5\HardLab)

链接：http://www.personal.soton.ac.uk/tb1m08/sparsify/sparsify_0_5.zip

function [s, err_mse, iter_time]=hard_l0_Mterm(x,A,m,M,varargin)
% hard_l0_Mterm: Hard thresholding algorithm that keeps exactly M elements
% in each iteration.
%
% This algorithm has certain performance guarantees as described in [1],
% [2] and [3].
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Usage
%
%   [s, err_mse, iter_time]=hard_l0_Mterm(x,P,m,M,'option_name','option_value')
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Input
%
%   Mandatory:
%               x   Observation vector to be decomposed
%               P   Either:
%                       1) An nxm matrix (n must be dimension of x)
%                       2) A function handle (type "help function_format"
%                          for more information)
%                          Also requires specification of P_trans option.
%                       3) An object handle (type "help object_format" for
%                          more information)
%               m   length of s
%               M   non-zero elements to keep in each iteration
%
%   Possible additional options:
%   (specify as many as you want using 'option_name','option_value' pairs)
%   See below for explanation of options:
%__________________________________________________________________________
%   option_name    |     available option_values                | default
%--------------------------------------------------------------------------
%   stopTol        | number (see below)                         | 1e-16
%   P_trans        | function_handle (see below)                |
%   maxIter        | positive integer (see below)               | n^2
%   verbose        | true, false                                | false
%   start_val      | vector of length m                         | zeros
%   step_size      | number                                     | 0 (auto)
%
%   stopping criteria used : (OldRMS-NewRMS)/RMS(x) < stopTol
%
%   stopTol: Value for stopping criterion.
%
%   P_trans: If P is a function handle, then P_trans has to be specified and
%            must be a function handle.
%
%   maxIter: Maximum number of allowed iterations.
%
%   verbose: Logical value to allow algorithm progress to be displayed.
%
%   start_val: Allows algorithms to start from partial solution.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Outputs
%
%    s              Solution vector
%    err_mse        Vector containing mse of approximation error for each
%                   iteration
%    iter_time      Vector containing computation times for each iteration
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Description
%
%   Implements the M-sparse algorithm described in [1], [2] and [3].
%   This algorithm takes a gradient step and then thresholds to only retain
%   M non-zero elements. It allows the step-size to be calculated
%   automatically as described in [3] and is therefore now independent from
%   a rescaling of P.
%
%
% References
%   [1]  T. Blumensath and M.E. Davies, "Iterative Thresholding for Sparse
%        Approximations", submitted, 2007
%   [2]  T. Blumensath and M. Davies; "Iterative Hard Thresholding for
%        Compressed Sensing" to appear Applied and Computational Harmonic
%        Analysis
%   [3] T. Blumensath and M. Davies; "A modified Iterative Hard
%        Thresholding algorithm with guaranteed performance and stability"
%        in preparation (title may change)
% See Also
%   hard_l0_reg
%
% Copyright (c) 2007 Thomas Blumensath
%
% The University of Edinburgh
% Email: thomas.blumensath@ed.ac.uk
% Comments and bug reports welcome
%
% This file is part of sparsity Version 0.4
% Created: April 2007
% Modified January 2009
%
% Part of this toolbox was developed with the support of EPSRC Grant
% D000246/1
%
% Please read COPYRIGHT.m for terms and conditions.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                    Default values and initialisation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%[n1 n2]=size(x);
if n2 == 1n=n1;
elseif n1 == 1x=x';n=n2;
elseerror('x must be a vector.');
endsigsize     = x'*x/n;
oldERR      = sigsize;
err_mse     = [];
iter_time   = [];
STOPTOL     = 1e-16;
MAXITER     = n^2;
verbose     = false;
initial_given=0;
s_initial   = zeros(m,1);
MU          = 0;if verbosedisplay('Initialising...')
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                           Output variables
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%switch nargout case 3comp_err=true;comp_time=true;case 2 comp_err=true;comp_time=false;case 1comp_err=false;comp_time=false;case 0error('Please assign output variable.')        otherwiseerror('Too many output arguments specified')
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                       Look through options
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Put option into nice format
Options={};
OS=nargin-4;
c=1;
for i=1:OSif isa(varargin{i},'cell')CellSize=length(varargin{i});ThisCell=varargin{i};for j=1:CellSizeOptions{c}=ThisCell{j};c=c+1;endelseOptions{c}=varargin{i};c=c+1;end
end
OS=length(Options);
if rem(OS,2)error('Something is wrong with argument name and argument value pairs.')
end
for i=1:2:OSswitch Options{i}case {'stopTol'}if isa(Options{i+1},'numeric') ; STOPTOL     = Options{i+1};   else error('stopTol must be number. Exiting.'); endcase {'P_trans'} if isa(Options{i+1},'function_handle'); Pt = Options{i+1};   else error('P_trans must be function _handle. Exiting.'); endcase {'maxIter'}if isa(Options{i+1},'numeric'); MAXITER     = Options{i+1};             else error('maxIter must be a number. Exiting.'); endcase {'verbose'}if isa(Options{i+1},'logical'); verbose     = Options{i+1};   else error('verbose must be a logical. Exiting.'); end case {'start_val'}if isa(Options{i+1},'numeric') && length(Options{i+1}) == m ;s_initial     = Options{i+1};  initial_given=1;else error('start_val must be a vector of length m. Exiting.'); endcase {'step_size'}if isa(Options{i+1},'numeric') && (Options{i+1}) > 0 ;MU     = Options{i+1};   else error('Stepsize must be between a positive number. Exiting.'); endotherwiseerror('Unrecognised option. Exiting.') end
endif nargout >=2err_mse = zeros(MAXITER,1);
end
if nargout ==3iter_time = zeros(MAXITER,1);
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                        Make P and Pt functions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%if          isa(A,'float')      P =@(z) A*z;  Pt =@(z) A'*z;
elseif      isobject(A)         P =@(z) A*z;  Pt =@(z) A'*z;
elseif      isa(A,'function_handle') tryif          isa(Pt,'function_handle'); P=A;else        error('If P is a function handle, Pt also needs to be a function handle. Exiting.'); endcatch error('If P is a function handle, Pt needs to be specified. Exiting.'); end
else        error('P is of unsupported type. Use matrix, function_handle or object. Exiting.'); end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                        Do we start from zero or not?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%if initial_given ==1;if length(find(s_initial)) > Mdisplay('Initial vector has more than M non-zero elements. Keeping only M largest.')ends                   =   s_initial;[ssort sortind]     =   sort(abs(s),'descend');s(sortind(M+1:end)) =   0;Ps                  =   P(s);Residual            =   x-Ps;oldERR      = Residual'*Residual/n;
elses_initial   = zeros(m,1);Residual    = x;s           = s_initial;Ps          = zeros(n,1);oldERR      = sigsize;
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                 Random Check to see if dictionary norm is below 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%x_test=randn(m,1);x_test=x_test/norm(x_test);nP=norm(P(x_test));if abs(MU*nP)>1;display('WARNING! Algorithm likely to become unstable.')display('Use smaller step-size or || P ||_2 < 1.')end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                        Main algorithm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if verbosedisplay('Main iterations...')
end
tic
t=0;
done = 0;
iter=1;while ~doneif MU == 0%Calculate optimal step size and do line searcholds                =   s;oldPs               =   Ps;IND                 =   s~=0;d                   =   Pt(Residual);% If the current vector is zero, we take the largest elements in dif sum(IND)==0[dsort sortdind]    =   sort(abs(d),'descend');IND(sortdind(1:M))  =   1;    end  id                  =   (IND.*d);Pd                  =   P(id);mu                  =   id'*id/(Pd'*Pd);s                   =   olds + mu * d;[ssort sortind]     =   sort(abs(s),'descend');s(sortind(M+1:end)) =   0;Ps                  =   P(s);% Calculate step-size requirement omega               =   (norm(s-olds)/norm(Ps-oldPs))^2;% As long as the support changes and mu > omega, we decrease muwhile mu > (0.99)*omega && sum(xor(IND,s~=0))~=0 && sum(IND)~=0
%             display(['decreasing mu'])% We use a simple line search, halving mu in each stepmu                  =   mu/2;s                   =   olds + mu * d;[ssort sortind]     =   sort(abs(s),'descend');s(sortind(M+1:end)) =   0;Ps                  =   P(s);% Calculate step-size requirement omega               =   (norm(s-olds)/norm(Ps-oldPs))^2;endelse% Use fixed step sizes                   =   s + MU * Pt(Residual);[ssort sortind]     =   sort(abs(s),'descend');s(sortind(M+1:end)) =   0;Ps                  =   P(s);endResidual            =   x-Ps;ERR=Residual'*Residual/n;if comp_errerr_mse(iter)=ERR;endif comp_timeiter_time(iter)=toc;end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                        Are we done yet?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%if comp_err && iter >=2if ((err_mse(iter-1)-err_mse(iter))/sigsize<STOPTOL);if verbosedisplay(['Stopping. Approximation error changed less than ' num2str(STOPTOL)])enddone = 1; elseif verbose && toc-t>10display(sprintf('Iteration %i. --- %i mse change',iter ,(err_mse(iter-1)-err_mse(iter))/sigsize)) t=toc;endelseif ((oldERR - ERR)/sigsize < STOPTOL) && iter >=2;if verbosedisplay(['Stopping. Approximation error changed less than ' num2str(STOPTOL)])enddone = 1; elseif verbose && toc-t>10display(sprintf('Iteration %i. --- %i mse change',iter ,(oldERR - ERR)/sigsize)) t=toc;endend% Also stop if residual gets too small or maxIter reachedif comp_errif err_mse(iter)<1e-16display('Stopping. Exact signal representation found!')done=1;endelseif iter>1 if ERR<1e-16display('Stopping. Exact signal representation found!')done=1;endendif iter >= MAXITERdisplay('Stopping. Maximum number of iterations reached!')done = 1; end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                    If not done, take another round
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%if ~doneiter=iter+1; oldERR=ERR;        end
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                  Only return as many elements as iterations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%if nargout >=2err_mse = err_mse(1:iter);
end
if nargout ==3iter_time = iter_time(1:iter);
end
if verbosedisplay('Done')
end

第三个版本：

文件：Demo_CS_IHT.m（部分）

链接：http://www.pudn.com/downloads518/sourcecode/math/detail2151378.html

function hat_x=cs_iht(y,T_Mat,m)
% y=T_Mat*x, T_Mat is n-by-m
% y - measurements
% T_Mat - combination of random matrix and sparse representation basis
% m - size of the original signal
% the sparsity is length(y)/4hat_x_tp=zeros(m,1);         % initialization with the size of original
s=floor(length(y)/4);        % sparsity
u=0.5;                       % impact factor% T_Mat=T_Mat/sqrt(sum(sum(T_Mat.^2))); % normalizae the whole matrixfor times=1:sx_increase=T_Mat'*(y-T_Mat*hat_x_tp);hat_x=hat_x_tp+u*x_increase;[val,pos]=sort((hat_x),'descend');  % why? worse performance with abs()hat_x(pos(s+1:end))=0;   % thresholding, keeping the larges s elementshat_x_tp=hat_x;          % updateend

7、单次重构代码

%压缩感知重构算法测试

clear all;close all;clc;
M = 64;%观测值个数
N = 256;%信号x的长度
K = 10;%信号x的稀疏度
Index_K = randperm(N);
x = zeros(N,1);
x(Index_K(1:K)) = 5*randn(K,1);%x为K稀疏的，且位置是随机的
Psi = eye(N);%x本身是稀疏的，定义稀疏矩阵为单位阵x=Psi*theta
Phi = randn(M,N);%测量矩阵为高斯矩阵
Phi = orth(Phi')';
A = Phi * Psi;%传感矩阵
% sigma = 0.005;
% e = sigma*randn(M,1);
% y = Phi * x + e;%得到观测向量y
y = Phi * x;%得到观测向量y
%% 恢复重构信号x
tic
theta = IHT_Basic(y,A,K);
% theta = cs_iht(y,A,size(A,2));
% theta = hard_l0_Mterm(y,A,size(A,2),round(1.5*K),'verbose',true);
x_r = Psi * theta;% x=Psi * theta
toc
%% 绘图
figure;
plot(x_r,'k.-');%绘出x的恢复信号
hold on;
plot(x,'r');%绘出原信号x
hold off;
legend('Recovery','Original')
fprintf('\n恢复残差：');
norm(x_r-x)%恢复残差

这里就不给出重构结果了，给出仿真结论：本人编的IHT基本版能够正常工作，但偶尔会重构失败；第二个版本hard_l0_Mterm.m重构效果很好；第三个版本Demo_CS_IHT.m重构效果很差，估计是作者疑问（why? worse performance with abs()），没有加abs取绝对值的原因吧……

8、结束语

8.1有关算法的名字

值得注意的是，在文献【2】中将式(2.2)称为iterative hard-thresholding algorithm，而将式(3.2)称为M-sparse algorithm，在文献【1】中又将式(3.2)称为Iterative Hard Thresholding algorithm (IHTs)，一般简称IHT的较多，多余的s指的是s稀疏。可见算法的名称是也是一不断完善的过程啊……

8.2与GraDeS算法的关系

如果你学习过GraDeS算法（参见http://blog.csdn.net/jbb0523/article/details/52059296），然后再学习本算法，是不是有一种似曾相似的感觉？

没错，这两个算法的迭代公式几乎是一样的，尤其是文献【1】中的式(12)（如上图第二个红框）进一步拓展了该算法的定义。这个就跟CoSaMP与SP两个算法一样，在GraDeS的提出文献【5】中开始部分还提到了IHT，但后面就没提了，不知道作者是怎么看待这个问题的。如果非说二者有区别，那就是GraDeS的参数γ=1+δ_2s，且δ_2s<1/3。

所以，有想法得赶紧写成论文发表出来，否则被抢了先机那就……

8.3重构效果问题

另外，在GraDeS算法中提到该算法的重构效果不好，这里注意文献【2】中的一段话：

也就是说，IHT作者也意识到了该种算法的问题，并提出了两种应用策略(two strategies for asuccessful application of the methods)。

8.4Landweber迭代

在网上搜索“Landweber迭代”时找到了一段程序^[6]：

function [x,k]=Landweber(A,b,x0)
alfa=1/sum(diag(A*A'));
k=1;
L=200;
x=x0;
while k<Lx1=x;x=x+alfa*A'*(b-A*x);if norm(b-A*x)/norm(b)<0.005break;elseif norm(x1-x)/norm(x)<0.001break;endk=k+1;
end

注意该程序的迭代部分“x=x+alfa*A'*(b-A*x);”，除了多了一些alfa系数外，这跟IHT不是基本一样么？或者说与GraDeS有什么区别？

有关LandWeber迭代可参见文献：“Landweber L. An iteration formula for Fredholm integral equations of the first kind[J]. American journal of mathematics, 1951, 73(3): 615-624.”，此处不再多述。

8.5改进算法

作者后来又提出了两个关于IHT的改进算法，分别是RIHT(Normalized IHT)^[7]和AIHT(Accelerated IHT)^[8]。

提出RIHT主要是由于IHT有一些缺点^[7]：

新算法RIHT将会有如下优点：

之所以作者提供的软件包（第二个版本IHT）重构效果更好是由于最新版的hard_l0_Mterm.m (\sparsify_0_5\HardLab)程序中已经更新成了RIHT。

RIHT的算法流程如下：

将IHT改进为AIHT后会有如下优点^[8]：

值得注意的是，AIHT应该是一类算法的总称（虽然作者只阐述了两种实现策略），这个类似于FFT是所有DFT快速算法的总称：

8.6稀疏度对IHT的影响

自己可以试一下，IHT输入参数中的稀疏度并不是很关键，若实际稀疏度为K，则稀疏度这个输入参数只要不小于K就可以了，重构效果都挺不错的，比如第三个版本的IHT程序，作者直接将稀疏度定义为信号y长度的四分之一。

8.7作者去向？

细心的人会发现，文献【8】的暑名单位为剑桥大学(University of Oxford)，并不是作者主页所在的南安普敦大学(University of Southampton)，在文献【8】的最后南提到：

Previous position?难道作者跳到Oxford了？

9、参考文献

【1】Blumensath T, Davies M E.Iterative hard thresholding for compressed sensing[J]. Applied & Computational HarmonicAnalysis, 2008, 27(3):265-274. (Available at:http://www.sciencedirect.com/science/article/pii/S1063520309000384)

【2】Blumensath T, Davies M E.Iterative Thresholding for Sparse Approximations[J]. Journal of Fourier Analysis & Applications,2008, 14(5):629-654. (Available at:http://link.springer.com/article/10.1007%2Fs00041-008-9035-z)

【3】Homepageof Blumensath T :http://www.personal.soton.ac.uk/tb1m08/index.html

【4】Lange, K., Hunter, D.R., Yang, I.. OptimizationTransfer Using Surrogate Objective Functions[J]. Journal of Computational &Graphical Statistics, 2000, 9(1):1-20. (Available at: http://sites.stat.psu.edu/~dhunter/papers/ot.pdf)

【5】GargR, Khandekar R. Gradient descent with sparsification: an iterative algorithmfor sparse recovery with restricted isometry property[C]//Proceedings of the26th Annual InternationalConference on Machine Learning. ACM, 2009: 337-344

【6】shasying2. landweber迭代方法.http://download.csdn.net/detail/shasying2/5092828

【7】Blumensath T, Davies M E.Normalized Iterative Hard Thresholding: Guaranteed Stability and Performance[J]. IEEE Journal of Selected Topics in Signal Processing, 2010,4(2):298-309.

【8】Blumensath T. Accelerated iterative hard thresholding[J]. Signal Processing, 2012, 92(3):752-756.

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