CLC-Conference-Proceeding-2018

xix or strategies based on the SVD of the known
data matrix xx .
A very general scheme of a solution
strategy would be:
Model selection and cost function.
Determination of the internal rank or dimension.
Initialization of the matrices according to 1.
Choose a solution method.
Estimation of the approximation error.
, which also refers to the Euclidean
distance xxi . This measure is the most simply and
frequently used. The Frobenius norm,
written as
the columns of A.
, can be equivalently
, with aj
1.2 Algorithmic ideas for the solution of NMF
The general classical problem to be solved as it
appears in 1.1 is:
Known Y ∈ R IxT + the data and a rank J, J ≤
min(I, T), find A = [a1,a2, … , aJ ] ∈ R IxJ + , and X
= B T =
T JxT T + E where E ∈ R IxT + represents the
approximation error [b1,b2,… , bJ ] ∈ R+ ,Y = AX
+ E = AB
This is a nonlinear optimization problem with
unknowns A and X and nonnegative constraints
(inequality constraints). It is not always possible
to find the matrices that exactly factor Y nor to
know the error matrix E. In general, the problem
of finding two matrices A and X is raised. That
minimizes a measure of
"distance" (according to the function of cost that
is used) between the data matrix and the product
AX.
The approximate problem of NMF, Y ≈ AX,
corresponds to minimizing some measure of
distance, divergence or measure of dissimilarity.
This could be formulated as, find A and X, such
that they satisfy
, where ‖ ‖F
represents the Frobenius norm, AIxJ, XJxT and A ≥
0, X ≥ 0 or as it is usually denoted DF(Y‖AX) =
Factors A and X are sought in other matrix
spaces, in particular in and in . For this
reason, the determination of the rank has a
special influence on the solution, since it
determines the search space. It is important to
emphasize that the function to be minimized
(objective or cost function) is convex with
respect to the elements of matrix A or with
respect to the elements of matrix X but not with
respect to both at the same time.
Another widely used measure is the generalized
Kullback-Leibler divergence (I-divergence):
From
p ln(p) ≥ p − 1 follows that the I- Divergence
takes non-negative values and it vanish if and
only if p =
q.
As a side effect, very often, the matrices A and X
are sparse. An interesting idea could be explicitly
control the sparsity of the matrices. In xxii some
sparseness criteria are presented that can be
imposed as constraints in the model.
Three main strategies are used to obtain the
matrices in the NMF.
Alternating Least Squares (ALS).
Descent methods.
Updating rules.