CLC-Conference-Proceeding-2018

solution is unique for any vector y and can be
obtained in polynomial time.
Another kind of methods can be used to
solve NMF, the descent algorithms. These
algorithms choose a descent direction to move
looking for a next solution that successively
improves the values of the objective function.
The methods of the projected gradient are
descent methods. This algorithms update the
approximation to the solution x k building the
x k +1
as follows:
x k +1
= x k + s k (̅x̅k̅
− x k ), where ̅x̅k̅ = P[x k − α k p k ]
here P[ ] denotes the projection in the set of
solutions, s k ∈ (0,1] is a step size, α k is a positive
scalar and p k is the direction of descent. Using
the ideas several methods can be constructed
taking the appropriate descent directions and the
rules to obtain the step size. For more
information on this type of methods see xxx .
The Updating rules can be obtained
considering an objective function and trying to
solve the problem xxxi .
For example, if the objective function is
minimize the Euclidean norm between Y and AX,
i.e.
min‖Y − AX‖ 2 = min∑ yij − (AX| ij) 2
ij
and the rules that derive from that are:
1.2 Initialization of the NMF
The initialization of the matrices in the
NMF have a direct influence in the solution and
that is why the solutions depend strongly on that.
Bad initializations produce slow convergence.
As already explained the objective function of
the ALS is not convex in both variables although
it is strictly convex in one variable, for example
in A or X and the algorithm can stop in a local
minimum.
The problem of initialization can be even
more complex if there is a particular structure of
the factorization, for example, in the symNMF or
the triNMF or if the matrix is large.
In xxxii it is proposed to follow the
following procedure:
Build a search algorithm in which the
best initialization is found in a space of R pairs
of matrices. Frequently the beginning of this
procedure can be with randomly generated
matrices or take them as the output of a simple
ALS algorithm. The R parameter usually
constitutes the necessary iterations in which 10
to 20 are considered enough.
Run a specific NMF algorithm for each
pair of matrices with a fixed number of iterations
(10 to 20 are also considered enough). As a
result, we obtain R pairs
initial matrices.
, from estimates of
Select the pair A Rmin ,X Rmin , that is, those with the
lowest evaluation of the objective function as
initialization for the factorization.
In xxxiii an implementation of this
procedure is presented, the Algorithm 1.2. To
read about the use of genetic algorithms used to
perform the initialization in NMF, see xxxiv
and xxxv .
1.3 Stopping criteria
Different stopping criteria can be considered for
the algorithms used to calculate the NMF. Some
of them are listed: