function [cost, grad] = softmaxCost(theta, numClasses, inputSize, lambda, data, labels)% numClasses - the number of classes % inputSize - the size N of the input vector% lambda - weight decay parameter% data - the N x M input matrix, where each column data(:, i) corresponds to%        a single test set% labels - an M x 1 matrix containing the labels corresponding for the input data%% Unroll the parameters from thetatheta = reshape(theta, numClasses, inputSize);numCases = size(data, 2);groundTruth = full(sparse(labels, 1:numCases, 1));%cost = 0;%thetagrad = zeros(numClasses, inputSize);%% ---------- YOUR CODE HERE --------------------------------------%  Instructions: Compute the cost and gradient for softmax regression.%                You need to compute thetagrad and cost.%                The groundTruth matrix might come in handy.weightDecay = (1/2) * lambda * sum(sum(theta.*theta));% M1(r, c) is theta(r)‘ * x(c)M1 = theta * data;% preventing overflowsM1 = bsxfun(@minus, M1, max(M1, [], 1));% M2(r, c) is exp(theta(r)‘ * x(c))M2 = exp(M1);% M2 is the predicted matrixM2 = bsxfun(@rdivide, M2, sum(M2));% 1{·} operator only preserve a part of positions of log(M2)M = groundTruth .* log(M2);cost = -(1/numCases) * sum(sum(M)) + weightDecay;% thetagradthetagrad = zeros(numClasses, inputSize);% difference between ground truth and predict valuediff = groundTruth - M2;for i=1:numClasses    thetagrad(i,:) = -(1/numCases) * sum((data .* repmat(diff(i,:), inputSize, 1)) ,2)‘ + lambda * theta(i,:);end% ------------------------------------------------------------------% Unroll the gradient matrices into a vector for minFuncgrad = thetagrad(:);end

function [pred] = softmaxPredict(softmaxModel, data)% softmaxModel - model trained using softmaxTrain% data - the N x M input matrix, where each column data(:, i) corresponds to%        a single test set%% Your code should produce the prediction matrix % pred, where pred(i) is argmax_c P(y(c) | x(i)). % Unroll the parameters from thetatheta = softmaxModel.optTheta;  % this provides a numClasses x inputSize matrix%pred = zeros(1, size(data, 2));%% ---------- YOUR CODE HERE --------------------------------------%  Instructions: Compute pred using theta assuming that the labels start %                from 1.result = theta * data;% sort by column[~,ind] = sort(result);pred = ind(size(theta,1), :);% ---------------------------------------------------------------------end