Function File: [W, H] = nmf_pg (V, Winit, Hinit, tol, timelimit, maxiter)

Non-negative matrix factorization by alternative non-negative least squares using projected gradients.

The matrix V is factorized into two possitive matrices W and H such that V = W*H + U. Where U is a matrix of residuals that can be negative or positive. When the matrix V is positive the order of the elements in U is bounded by the optional named argument tol (default value 1e-9).

The factorization is not unique and depends on the inital guess for the matrices W and H. You can pass this initalizations using the optional named arguments Winit and Hinit.

timelimit, maxiter: limit of time and iterations

Examples: 

  A     = rand(10,5);
  [W H] = nmf_pg(A,tol=1e-3);
  U     = W*H -A;
  disp(max(abs(U)));

Demonstration 1

The following code

 t = linspace (0,1,100)';

 ## --- Build hump functions of different frequency
 W_true = arrayfun ( @(f)sin(2*pi*f*t).^2, linspace (0.5,2,4), ...
                                                     'uniformoutput', false );
 W_true = cell2mat (W_true);
 ## --- Build combinator vectors
 c      = (1:4)';
 H_true = arrayfun ( @(f)circshift(c,f), linspace (0,3,4), ...
                                                     'uniformoutput', false );
 H_true = cell2mat (H_true);
 ## --- Mix them
 V = W_true*H_true;
 ## --- Give good inital guesses
 Winit = W_true + 0.4*randn(size(W_true));
 Hinit = H_true + 0.2*randn(size(H_true));
 ## --- Factorize
 [W H] = nmf_pg(V,'Winit',Winit,'Hinit',Hinit,'Tol',1e-6,'MaxIter',1e3);
 disp('True mixer')
 disp(H_true)
 disp('Rounded factorized mixer')
 disp(round(H))
 ## --- Plot results
 plot(t,W,'o;factorized;')
 hold on
 plot(t,W_true,'-;True;')
 hold off
 axis tight

Produces the following output

--- Factorizing 100-by-4 matrix into 100-by-4 times 4-by-4
Initial gradient norm = 448.013653
Running main loop, this may take a while.

Iterations = 8
Final proj-grad norm = 0.000256
True mixer
   1   4   3   2
   2   1   4   3
   3   2   1   4
   4   3   2   1
Rounded factorized mixer
   1   4   3   2
   2   1   4   3
   3   2   1   4
   4   3   2   1

and the following figure

Figure 1

Package: linear-algebra