CLC-Conference-Proceeding-2018

e obtained as the sum of matrices of rank 1,
ajbj T . If the decomposition is exact (E = 0), then
it is named Non-negative Rank Factorization
(NRF) xii . The rank J or internal dimension is also
called a nonnegative rank and is denoted by
rank+(Y) xiii . The rank J satisfies rank(Y) ≤
rank+(Y) ≤ min{I, T}.
In general, the rank J usually satisfies
.
Figure 1: Classic model of NMF
In Figure 1 taken from xiv , page 9, the structure of
the factorization can be observed.
It is not difficult to notice that an NMF does not
have to be an NRF. Non-negative matrices
cannot always be found such that factorize the Y
matrix in an exactly way or that the matrix E = 0.
This is due, among others, to the approximation
errors (rounding) that are generated and
propagated in the calculations in a finite
precision arithmetic.
To start the factorization process three
fundamental questions must be answered. The
first is related to the NMF model that will be
used, this depends on the characteristics of the
problem being solved or the type of the data.
NMF have been adapted to many applications.
Choosing the model involves selecting
the cost function to be minimized and the
additional constraints to be imposed to the
matrices, as we will see later. In this work we
present some of the applications in which the
Images Group of the University of Havana is
currently working and the models that are being
studied.
The second question is related to the
dimension or rank of the matrix J, J ≤ min(I, T),
or also known as internal dimension. Of course,
this choice has a lot to do with the a priori
knowledge of the problem to be treated. NMF
are considered methods of dimensionality
reduction so that this choice is crucial both for
the appropriate modeling of the problem and for
obtaining the solutions. The rank should be set to
address the solution of the problem. Several
approaches to determine the rank of the matrix
appear in the literature. Several authors prefer
the use the SVD xv . Another approaches suggest
to obtain the graph of the singular values and
select the range after observing when the graph
has an elbow. This fact is related to the theory of
discrete ill-posed problems and the regularization
of solutions that is very well studied in xvi . The
Images Team of the University of Havana is
studying a criterion proposed in xvii to compare
the solutions with the use of the SVD. Other
authors have agreed that a suitable variant is the
use of heuristics based on populations as in xviii .
The third question is related to the
initialization of the matrices depending on the
model to be used. It has been shown that the
algorithms that are used to find the matrices are
sensitive to the initialization. The convergence
and the speed of convergence of the algorithms
may be affected by the initialization and with
this, of course, the quality of the solutions. To
answer this question we could also cite many
works. The most intuitive ideas are random
initializations. These in general do not take into
account the characteristics of the problem. Other
works propose to use population-based heuristics