rank_1_factorization.m

function [bg_spatial, bg_temporal] = rank_1_factorization(Y, maxIter)
%% RANK_1_FACTORIZATION  Rank-1-matrix-factorization of the movie Y
%       Y ~ bg_spatial * bg_temporal
%
% Input:
% Y                      movie
% max_iter               maximum Number of Iterations
%
% Output:
% bg_temporal            temporal component of the rank-1-factorization
% bg_spatial             spatial component of the rank-1-factorization
%
% This algorithm performs a form a block-wise gradient descent on the objective function
%
% $$L \left( s, t \right) = \sum_{i j}^{} \left( Y_{i j} - s_i t_j \right) ^2$$
% 
% Here, s corresponds to *bg_spatial*, and t corresponds to *bg_temporal*.
% This can be seen by calculating the gradient along s and t
% 
% $$D_s L=2\left( \|s\|_2^2\,s - s\,Y\right)$$
% 
% $$D_t L=2\left( \|t\|_2^2\,t - t\,Y\right)$$
%
% and set them to zero. Between update we normalize the previously updated component. 
% This simplifies the code and leads to better performance in the general case of NNMF.

if nargin<2
    maxIter=1;
end

bg_spatial = ones(size(Y,1),1)/sqrt(size(Y,1));
for iter=1:maxIter
    bg_temporal = bg_spatial'*Y;
    bg_temporal = bg_temporal/norm(bg_temporal(:));
    bg_spatial = Y*bg_temporal';
    if iter<maxIter
        bg_spatial = bg_spatial/norm(bg_spatial(:));
    end
end
end