SVD在稀疏表示中的应用

SVD:singular value decomposition奇异值分解

在认识SVD之前，先来学习两个相关的概念：正交矩阵和酉矩阵。

如果，则n阶实矩阵A称为正交矩阵。而酉矩阵是正交矩阵往复数域上的推广。
判断正交矩阵和酉矩阵的充分必要条件是：。或者说正交矩阵和酉矩阵的共轭转置和它的逆矩阵相等。

对任意矩阵，都能被奇异值分解为

其中是的正交矩阵，是的正交矩阵，是由r个沿对角线从大到小排列的奇异值组成的方阵，就是矩阵的秩。奇异值分解是一种正交矩阵分解法。

奇异值分解是根据方阵的特征值分解推导而来,特征值的几何意义,在图像处理中,上面等式的左边方阵可以看成一个变换矩阵,x看成是进空间中的
一个点,那么几何意义就是,经过该几何变换后,该点还是存在于过原点与该点的直线的上.
所以说一个方阵的特征向量就是这样一个向量,经过这种特定的变换后保持方向不变,只是长度上的伸缩而已

而在稀疏表示中使用K-SVD中更新字典原子,具体为:
1 第一步是初始化系数矩阵,而在更新系数矩阵之前,需要将字典进行初始化,只有得到字典后,才能够求稀疏表示系数,字典初始化方法有:1 直接从训练样本中挑选一些向量用于构建字典,也可以自己定义一
些方法对字典进行初始化,字典初始化后,就可以通过OMP算法,正交基追踪算法求解稀疏表示系数,当然,还有其他许多方法可以进行求解

2 对字典中的各个原子进行更新:
更新的算法还是从公式进行入手:

其中Y是训练样本,D是前期已经初始化的字典,X是根据初始化的字典求解的稀疏表示系数,dk是待更新的原子,xt为系数矩阵中与dk相乘的行,Y是mxn,m是行数,也是Y中一个样本向量的维度,n是样本的个数,D是字典,mxa,m是原子的维度,a是字典中原子的个数,X是axn,a是代表的是X的行数,n代表的是X的列数,X中的每一列代表的是对应Y中对应列样本的在字典D中选择的各个原子的系数,其中

最后得到上面的公式后,目标就是要该值最小,该列原子向量以及对应的系数是未知的,那么直接对Ek进行SVD分解
将得到的对应的特征值最大的特征向量作为字典的该列原子,将特征值作为系数进行,更新
这样就更新了一个原子和稀疏表示系数,重复这个步骤,直到所有原子都更新完,这样更新完得到的原子,实现了所谓的稀疏表示

需要注意的是,如果在上面的公式中直接用SVD进行更新,SVD能够找到距离Ek最近的秩为1的矩阵,单张这样得到的系数xt不稀疏,换句话说,xt与更新xt的非零元所处位置和value不一样,直观的解决办法是只保留稀疏中的非零值,由于系数在前面求解时已经实现稀疏性,因此只保留非零项,再进行SVD分解就会保留xt的稀疏解,对应的matlab代码中也有体现:

function [betterDictionaryElement,CoefMatrix,NewVectorAdded] = I_findBetterDictionaryElement(Data,Dictionary,j,CoefMatrix,numCoefUsed)if (length(who('numCoefUsed'))==0)    numCoefUsed = 1;end%只取稀疏矩阵中与j列原子相关的,且非零的系数的位置%Dictionary为MxA,CoefMatrix为AxN,与Dictionary中第j列相关的只是CoefMatrix中的第j行,因为只有CoefMatrix中的%第j行才会与Dictionary中的第j列的元素相乘,才能够有交集%同时这里只取非零的值,是为了保持稀疏性relevantDataIndices = find(CoefMatrix(j,:)); % the data indices that uses the j'th dictionary element.if (length(relevantDataIndices)<1) %(length(relevantDataIndices)==0)    ErrorMat = Data-Dictionary*CoefMatrix;    ErrorNormVec = sum(ErrorMat.^2);    [d,i] = max(ErrorNormVec);    betterDictionaryElement = Data(:,i);%ErrorMat(:,i); %    betterDictionaryElement = betterDictionaryElement./sqrt(betterDictionaryElement'*betterDictionaryElement);    betterDictionaryElement = betterDictionaryElement.*sign(betterDictionaryElement(1));    CoefMatrix(j,:) = 0;    NewVectorAdded = 1;    return;endNewVectorAdded = 0;%取tmpCoefMatrix中的所有relevantDataIndices对应的列,列的个数就是数组relevantDataIndices的长度%tmpCoefMatrix(j,:)置0是为因为为了求解,所以要置0%Data(:,relevantDataIndices)只取data对应的relevantDataIndices的列,是因为只有这些列是与Dictionary第%j列原子相关,因此tmpCoefMatrix = CoefMatrix(:,relevantDataIndices); tmpCoefMatrix(j,:) = 0;% the coeffitients of the element we now improve are not relevant.%这里的errors就是上面的Ek,稍有不同的是,这里只选取了Y中与字典中第j个原子相关的列errors =(Data(:,relevantDataIndices) - Dictionary*tmpCoefMatrix); % vector of errors that we want to minimize with the new element% % the better dictionary element and the values of beta are found using svd.% % This is because we would like to minimize || errors - beta*element ||_F^2. % % that is, to approximate the matrix 'errors' with a one-rank matrix. This% % is done using the largest singular value.%svds(errors,1)的意思是对errors进行奇异值分解,分解后只保留对应特征值最大的左边向量,特征值,右边向量%如果参数1变成2就是保留左边特征值最大的前2个左边向量,右边向量和前2个特征值,依次类推%svds分解得到的是对应的维度为Ax1的向量:betterDictionaryElement%以及1x1的特征值:singularValue,维度为errors的列x1的向量:betaVector%singularValue*betaVector'其实也就是对新求得到的原子betterDictionaryElement%上relevantDataIndies位置处元素的系数,用公式表示就是:%errors = betterDictionaryElement*singularValue*betaVector'%各个向量大小从左到右依次是AxB = Ax1 * 1*1 * (B*1)'%从上面这行公式也不难看出,这里所谓的稀疏性,就是通过前面OMP只取非零项%以及这里只取特征向量最大的值对应的行向量,列向量来实现[betterDictionaryElement,singularValue,betaVector] = svds(errors,1);CoefMatrix(j,relevantDataIndices) = singularValue*betaVector';% *signOfFirstElem

然后再重复上面的步骤m次,将字典每个原子迭代更新m次,直到迭代完成,训练数据Y对应的原子和稀疏表示系数就得到了;完整的ksvd代码:

function [Dictionary,output] = KSVD(...    Data,... % an nXN matrix that contins N signals (Y), each of dimension n.    param)% =========================================================================%                          K-SVD algorithm% =========================================================================% The K-SVD algorithm finds a dictionary for linear representation of% signals. Given a set of signals, it searches for the best dictionary that% can sparsely represent each signal. Detailed discussion on the algorithm% and possible applications can be found in "The K-SVD: An Algorithm for % Designing of Overcomplete Dictionaries for Sparse Representation", written% by M. Aharon, M. Elad, and A.M. Bruckstein and appeared in the IEEE Trans. % On Signal Processing, Vol. 54, no. 11, pp. 4311-4322, November 2006. % =========================================================================% INPUT ARGUMENTS:% Data                         an nXN matrix that contins N signals (Y), each of dimension n. % param                        structure that includes all required%                                 parameters for the K-SVD execution.%                                 Required fields are:%    K, ...                    the number of dictionary elements to train%    numIteration,...          number of iterations to perform.%    errorFlag...              if =0, a fix number of coefficients is%                                 used for representation of each signal. If so, param.L must be%                                 specified as the number of representing atom. if =1, arbitrary number%                                 of atoms represent each signal, until a specific representation error%                                 is reached. If so, param.errorGoal must be specified as the allowed%                                 error.%    preserveDCAtom...         if =1 then the first atom in the dictionary%                                 is set to be constant, and does not ever change. This%                                 might be useful for working with natural%                                 images (in this case, only param.K-1%                                 atoms are trained).%    (optional, see errorFlag) L,...                 % maximum coefficients to use in OMP coefficient calculations.%    (optional, see errorFlag) errorGoal, ...        % allowed representation error in representing each signal.%    InitializationMethod,...  mehtod to initialize the dictionary, can%                                 be one of the following arguments: %                                 * 'DataElements' (initialization by the signals themselves), or: %                                 * 'GivenMatrix' (initialization by a given matrix param.initialDictionary).%    (optional, see InitializationMethod) initialDictionary,...      % if the initialization method %                                 is 'GivenMatrix', this is the matrix that will be used.%    (optional) TrueDictionary, ...        % if specified, in each%                                 iteration the difference between this dictionary and the trained one%                                 is measured and displayed.%    displayProgress, ...      if =1 progress information is displyed. If param.errorFlag==0, %                                 the average repersentation error (RMSE) is displayed, while if %                                 param.errorFlag==1, the average number of required coefficients for %                                 representation of each signal is displayed.% =========================================================================% OUTPUT ARGUMENTS:%  Dictionary                  The extracted dictionary of size nX(param.K).%  output                      Struct that contains information about the current run. It may include the following fields:%    CoefMatrix                  The final coefficients matrix (it should hold that Data equals approximately Dictionary*output.CoefMatrix.%    ratio                       If the true dictionary was defined (in%                                synthetic experiments), this parameter holds a vector of length%                                param.numIteration that includes the detection ratios in each%                                iteration).%    totalerr                    The total representation error after each%                                iteration (defined only if%                                param.displayProgress=1 and%                                param.errorFlag = 0)%    numCoef                     A vector of length param.numIteration that%                                include the average number of coefficients required for representation%                                of each signal (in each iteration) (defined only if%                                param.displayProgress=1 and%                                param.errorFlag = 1)% =========================================================================if (~isfield(param,'displayProgress'))    param.displayProgress = 0;endtotalerr(1) = 99999;if (isfield(param,'errorFlag')==0)    param.errorFlag = 0;endif (isfield(param,'TrueDictionary'))    displayErrorWithTrueDictionary = 1;    ErrorBetweenDictionaries = zeros(param.numIteration+1,1);    ratio = zeros(param.numIteration+1,1);else    displayErrorWithTrueDictionary = 0;    ratio = 0;endif (param.preserveDCAtom>0)    FixedDictionaryElement(1:size(Data,1),1) = 1/sqrt(size(Data,1));else    FixedDictionaryElement = [];end% coefficient calculation method is OMP with fixed number of coefficientsif (size(Data,2) < param.K)    disp('Size of data is smaller than the dictionary size. Trivial solution...');    Dictionary = Data(:,1:size(Data,2));    return;elseif (strcmp(param.InitializationMethod,'DataElements'))    Dictionary(:,1:param.K-param.preserveDCAtom) = Data(:,1:param.K-param.preserveDCAtom);elseif (strcmp(param.InitializationMethod,'GivenMatrix'))    Dictionary(:,1:param.K-param.preserveDCAtom) = param.initialDictionary(:,1:param.K-param.preserveDCAtom);end% reduce the components in Dictionary that are spanned by the fixed% elementsif (param.preserveDCAtom)    tmpMat = FixedDictionaryElement \ Dictionary;    Dictionary = Dictionary - FixedDictionaryElement*tmpMat;end%normalize the dictionary.Dictionary = Dictionary*diag(1./sqrt(sum(Dictionary.*Dictionary)));Dictionary = Dictionary.*repmat(sign(Dictionary(1,:)),size(Dictionary,1),1); % multiply in the sign of the first element.totalErr = zeros(1,param.numIteration);% the K-SVD algorithm starts here.for iterNum = 1:param.numIteration    % find the coefficients    if (param.errorFlag==0)        %CoefMatrix = mexOMPIterative2(Data, [FixedDictionaryElement,Dictionary],param.L);        CoefMatrix = OMP([FixedDictionaryElement,Dictionary],Data, param.L);    else         %CoefMatrix = mexOMPerrIterative(Data, [FixedDictionaryElement,Dictionary],param.errorGoal);        CoefMatrix = OMPerr([FixedDictionaryElement,Dictionary],Data, param.errorGoal);        param.L = 1;    end    replacedVectorCounter = 0;    rPerm = randperm(size(Dictionary,2));    for j = rPerm        [betterDictionaryElement,CoefMatrix,addedNewVector] = I_findBetterDictionaryElement(Data,...            [FixedDictionaryElement,Dictionary],j+size(FixedDictionaryElement,2),...            CoefMatrix ,param.L);        Dictionary(:,j) = betterDictionaryElement;        if (param.preserveDCAtom)            tmpCoef = FixedDictionaryElement\betterDictionaryElement;            Dictionary(:,j) = betterDictionaryElement - FixedDictionaryElement*tmpCoef;            Dictionary(:,j) = Dictionary(:,j)./sqrt(Dictionary(:,j)'*Dictionary(:,j));        end        replacedVectorCounter = replacedVectorCounter+addedNewVector;    end    if (iterNum>1 & param.displayProgress)        if (param.errorFlag==0)            output.totalerr(iterNum-1) = sqrt(sum(sum((Data-[FixedDictionaryElement,Dictionary]*CoefMatrix).^2))/prod(size(Data)));            disp(['Iteration   ',num2str(iterNum),'   Total error is: ',num2str(output.totalerr(iterNum-1))]);        else            output.numCoef(iterNum-1) = length(find(CoefMatrix))/size(Data,2);            disp(['Iteration   ',num2str(iterNum),'   Average number of coefficients: ',num2str(output.numCoef(iterNum-1))]);        end    end    if (displayErrorWithTrueDictionary )         [ratio(iterNum+1),ErrorBetweenDictionaries(iterNum+1)] = I_findDistanseBetweenDictionaries(param.TrueDictionary,Dictionary);        disp(strcat(['Iteration  ', num2str(iterNum),' ratio of restored elements: ',num2str(ratio(iterNum+1))]));        output.ratio = ratio;    end    Dictionary = I_clearDictionary(Dictionary,CoefMatrix(size(FixedDictionaryElement,2)+1:end,:),Data);    if (isfield(param,'waitBarHandle'))        waitbar(iterNum/param.counterForWaitBar);    endendoutput.CoefMatrix = CoefMatrix;Dictionary = [FixedDictionaryElement,Dictionary];%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  findBetterDictionaryElement%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [betterDictionaryElement,CoefMatrix,NewVectorAdded] = I_findBetterDictionaryElement(Data,Dictionary,j,CoefMatrix,numCoefUsed)if (length(who('numCoefUsed'))==0)    numCoefUsed = 1;endrelevantDataIndices = find(CoefMatrix(j,:)); % the data indices that uses the j'th dictionary element.if (length(relevantDataIndices)<1) %(length(relevantDataIndices)==0)    ErrorMat = Data-Dictionary*CoefMatrix;    ErrorNormVec = sum(ErrorMat.^2);    [d,i] = max(ErrorNormVec);    betterDictionaryElement = Data(:,i);%ErrorMat(:,i); %    betterDictionaryElement = betterDictionaryElement./sqrt(betterDictionaryElement'*betterDictionaryElement);    betterDictionaryElement = betterDictionaryElement.*sign(betterDictionaryElement(1));    CoefMatrix(j,:) = 0;    NewVectorAdded = 1;    return;endNewVectorAdded = 0;tmpCoefMatrix = CoefMatrix(:,relevantDataIndices); tmpCoefMatrix(j,:) = 0;% the coeffitients of the element we now improve are not relevant.errors =(Data(:,relevantDataIndices) - Dictionary*tmpCoefMatrix); % vector of errors that we want to minimize with the new element% % the better dictionary element and the values of beta are found using svd.% % This is because we would like to minimize || errors - beta*element ||_F^2. % % that is, to approximate the matrix 'errors' with a one-rank matrix. This% % is done using the largest singular value.[betterDictionaryElement,singularValue,betaVector] = svds(errors,1);CoefMatrix(j,relevantDataIndices) = singularValue*betaVector';% *signOfFirstElem%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  findDistanseBetweenDictionaries%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [ratio,totalDistances] = I_findDistanseBetweenDictionaries(original,new)% first, all the column in oiginal starts with positive values.catchCounter = 0;totalDistances = 0;for i = 1:size(new,2)    new(:,i) = sign(new(1,i))*new(:,i);endfor i = 1:size(original,2)    d = sign(original(1,i))*original(:,i);    distances =sum ( (new-repmat(d,1,size(new,2))).^2);    [minValue,index] = min(distances);    errorOfElement = 1-abs(new(:,index)'*d);    totalDistances = totalDistances+errorOfElement;    catchCounter = catchCounter+(errorOfElement<0.01);endratio = 100*catchCounter/size(original,2);%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  I_clearDictionary%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function Dictionary = I_clearDictionary(Dictionary,CoefMatrix,Data)T2 = 0.99;T1 = 3;K=size(Dictionary,2);Er=sum((Data-Dictionary*CoefMatrix).^2,1); % remove identical atomsG=Dictionary'*Dictionary; G = G-diag(diag(G));for jj=1:1:K,    if max(G(jj,:))>T2 | length(find(abs(CoefMatrix(jj,:))>1e-7))<=T1 ,        [val,pos]=max(Er);        Er(pos(1))=0;        Dictionary(:,jj)=Data(:,pos(1))/norm(Data(:,pos(1)));        G=Dictionary'*Dictionary; G = G-diag(diag(G));    end;end;