ADMM求解优化NMF

本文拟对文章：“An Alternating Direction Algorithm for Matrix Completion with Nonnegative Factors”中利用ADMM进行非负矩阵分解部分进行推导。

他的 augmented Lagrangian 可写成如下形式：

(1)
(1)优化W

(2)优化H:

（3）优化Wp,Hp:

(4)优化:

迭代终止条件：

orl人脸库（32*32）上，目标函数收敛曲线：

 聚类结果比较：

略好于传统NMF优化方法基于原始数据的聚类结果。

matlab代码：

function [W, H] = nmf_admm(V, W, H)
% nmf_admm(V, W, H)

%
% inputs
%    V: matrix to factor 
%    W, H: initializations for W and H
%
% outputs
%    W, H: factorization such that V \approx W*H

    % determine dimensions
    [m,n] = size(V);
    [~,k] = size(W);
	rho = 1;
    gamma = 0.01;
    % initializations for other variables
    X = V';
    Wplus = W;
    Hplus = H;
    alphaW = zeros(size(W));
    alphaH = zeros(size(H));
    maxiter = 10000;
    for iter=1:maxiter
        % update for W
        P = H*H' + rho*eye(k);
        Q = X*H' + rho*Wplus - alphaW;
        W = ( P'\ Q' )';

        % update for H
        H = (W'*W + rho*eye(k)) \ (W'*X- alphaH + rho*Hplus);
        
        % update for H_+ and W_+
        Hplus = max((H + 1/rho*(alphaH)), 0);
        Wplus = max(W + 1/rho*alphaW, 0);

        % update for dual variables
        alphaH = alphaH + gamma*rho*(H - Hplus);
        alphaW = alphaW + gamma*rho*(W - Wplus);

        temp1 = 0.5*trace((X-W*H)'*(X-W*H));
        temp2 = trace(alphaW'*(W-Wplus)) + trace(alphaH'*(H-Hplus));
        temp3 = 0.5*rho*(trace((Wplus - W)*(Wplus - W)')+trace((Hplus - H)*(Hplus - H)'));

        fk = temp1 + temp2 + temp3;
        fk0 = fk/trace(X*X').^0.5;
        if iter>1
            err(iter-1) = fk;
           if abs(fk0-fk1)/max(1,abs(fk))<1e-6 
                break;
           end
        end
        fk1=fk0;
%         alphaW1 = alphaW;
%         alphaH1 = alphaH;
    end
    
    W = Wplus;
    H = Hplus; 

end