To obtain an NMF of a matrix A, the following problem arises:

Given the matrix A, with elements a(i,j) i=1,2,...,m and j=1,2,...,n , search matrix W(mxr), H(rxn) , both things to non-negative elements so that A=HH. As this equality in general can not be found in an exact way, the well-known classical model is proposed:

min(abs(a-WH)) subject to W>=0, H>=0 . In the literature the Frobenius and Euclidean norms are frequently presented, among other expressions. Other authors formulate the problem, for example, as A = WH + E. The problem of minimization is a problem of nonlinear optimization and must be solved with some of the methods or strategies appropriate for it.

This way of factoring matrices of non-negative elements allows us to find the factors at a scale commensurate with the problem being treated. If we compare the NMF with other factorizations such as the SVD, the QR, we see that matrices with positive and negative elements are searched. In the image processing, for example, the data are all non-negative, which does not allow an adequate interpretation.

This way of factoring a non-negative matrix allows us to introduce new questions to the problem in order to adapt the model better and better to the problem at hand.

The issues that influence the solution and the quality of the approach to the matrix are: the definition of the internal dimension, which is the factor of reduction of the model and the way to initialize the matrices in the solution methods.

The questions are:

1) How do you define the rank or internal dimension r which is the number of columns of W and the number of rows of H where A=WH?

2) Which initialization techniques best optimize the problem, using or not heuristics?