CLC-Conference-Proceeding-2018

Nonnegative matrix factorizations: Ideas and applications
Marta Lourdes Baguer Díaz-Romañach
This article is not intended to conduct an
exhaustive study of the state of the art on the
Non-Negative Matrix Factorizations (NMF) nor
to present all the possible extensions of its
general model or ways of solution, it is only
intended to motivate the reader to pay attention
to a versatile tool for scientific computing that
can be adapted to solve a wide range of problems
and that is still in development.
When one talk about the NMF, some
notes of Gene Golub take us back to the 70's i .
These factorizations shall be understood, for
many reasons, as a "philosophy" to address the
solution of different problems with nonnegative
data, taking into account the relationship
between the parts and the whole as seen by Lee
and Seung, ii more than a factorization in the
mathematical sense like, for example, the LU, the
QR or the SVD . This article will show some
aspects that are important to begin to get into the
study of NMF.
When the Linear Numerical Algebra is
explained, it begins by presenting the LU
factorization. This factorization looks for two
regular matrices, L, lower triangular, and U,
upper triangular, so that the system of linear
equations Ax = b, becomes easier to solve. In
this factorization the columns of the matrix A can
be expressed as a linear combination of the
columns of L.
That is, if we consider A = [a1, a2, … ,
an], L = [l1, l2,… , ln] and U = [u1, u2,… , un],
then, each column of the matrix A, can be written
as ai = u1il1 + u2il2 + ⋯ + uniln. In this way we
have obtained a basis L of the subspace
generated by the columns of the matrix A
without other requirements than the linear
independence of the columns. For ill
conditioned, rank deficient or simply rectangular
matrices, it is important to demand the
orthogonality of the basis. This is closely related
to the control of the propagation of the error. In
those cases, a QR factorization or some variant,
as the RRQR (rank revealing QR) can be
computed. iii
The Singular Value Decomposition
(SVD) is not simply a matrix factorization. The
SVD helps us to understand the main properties
of the matrix and finds application in
innumerable areas. Only the fact of being able to
handle in floating point arithmetic the concept of
the numerical rank of a matrix is an important
achievement. When obtaining a decomposition
of the matrix in singular values, orthonormal
basis are obtained both for the subspace
generated by the columns and for the subspace
generated by the rows of the matrix A ∈ R mxn iv .
A classic formulation is when we express
A = UΣV T , where the orthogonal matrices U ∈
R mxm and V ∈ R nxn contain the corresponding
basis and Σ ∈ R mxn , is a diagonal matrix and
contains the singular values σi, with σ1 ≥ σ2 ≥ ⋯
≥ σr > 0, and r is the rank of the matrix. An
important issue is that, it is not necessary to store
the complete U and V matrices but only the r
columns of the matrices U and V.
The matrix A can then be decomposed
into A = UrΣrVr T = Ar =
, a weighted
sum of matrices of rank 1, uivi T . It can be shown
that one of the main properties of the SVD is that
by making use of this expansion, we obtain the
best approximation of the rank k of A in