MOD 之"Hello World"
首先声明，MOD不是取模函数！MOD是字典学习和sparse coding的一种方法… 最近在看KSVD，其简化版就是MOD（method of directions）,这么说吧，KSVD和MOD的优化目标函数是相同的，MOD之所以可以称作KSVD的简化版是因为KSVD在MOD的基础上做了顺序更新列的优化。关于KSVD和MOD的理论知识请见下面我给出的一页note和referenc中的paper。本文主要给出其基本思想及我的代码，已经过测试，如有bug欢迎提出。
Reference
<<From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images>>, Page 68~70
KSVD & MOD's principle & objective function
Principle:
简单来说，其优化就是一个OMP（orthogonal matching pursuit）与Regression的迭代过程，因此代码包括一个OMP.m, regression.m.
Objective Function & the variation from MOD to KSVD:
Code
CODE1. MOD
运行Main（Main中通过MOD）学习字典和稀疏表示，MOD迭代调用Regression学习字典，调用和OMP获得sparse representation.
Main.m
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;关于Machine Learning更多的学习资料与相关讨论将继续更新，敬请关注本博客和新浪微博Rachel____Zhang.