This class implements the Nonsmooth Nonnegative Matrix Factorization (nsNMF) model, required by the Nonsmooth NMF algorithm.
The Nonsmooth NMF algorithm is defined by Pascual-Montano et al. (2006) as a modification of the standard divergence based NMF algorithm (see section Details and references below). It aims at obtaining sparser factor matrices, by the introduction of a smoothing matrix.
The Nonsmooth NMF algorithm is a modification of the standard divergence
based NMF algorithm (see NMF-class).
Given a non-negative n x p matrix V and a
factorization rank r, it fits the following model:
V ~ W S(theta) H, where:
W and H are such as in the standard model, i.e.
non-negative matrices of dimension n x r
and r x p respectively;
S is a r \times r square matrix whose entries depends on
an extra parameter 0\leq \theta \leq 1 in the following way:
S = (1-\theta)I + \frac{\theta}{r} 11^T ,
where I is the identity matrix and 1
is a vector of ones.
The interpretation of S as a smoothing matrix can be explained as follows:
Let X be a positive, nonzero, vector. Consider the transformed vector
Y = S X. If \theta = 0, then Y = X and no smoothing on
X has occurred. However, as theta tends to 1, the
vector Y tends to the constant vector with all elements almost equal
to the average of the elements of X. This is the smoothest possible
vector in the sense of non-sparseness because all entries are equal to the
same nonzero value, instead of having some values close to zero and others
clearly nonzero.
signature(object = "NMFns"): Compute estimate for an NMFns object, according to the Nonsmooth NMF model
(cf. NMFns-class).
Extra arguments in ... are passed to method smoothing, and are
typically used to pass a value for theta, which is used to compute
the smoothing matrix instead of the one stored in object.
signature(object = "NMFns"): Show method for objects of class NMFns
Object of class NMFns can be created using the standard way with
operator new
However, as for all NMF model classes -- that extend class
NMF-class, objects of class NMFns should be
created using factory method nmfModel :
new('NMFns')
nmfModel(model='NMFns')
nmfModel(model='NMFns', W=w, theta=0.3
See nmfModel for more details on how to use the factory
method.
The Nonsmooth NMF algorithm uses a modified version of the multiplicative
update equations in Lee & Seung's method for Kullback-Leibler divergence
minimization.
The update equations are modified to take into account the --
constant -- smoothing matrix.
The modification reduces to using matrix W S instead of matrix W
in the update of matrix H, and similarly using matrix S H
instead of matrix H in the update of matrix W.
After the matrix W has been updated, each of its columns is scaled so
that it sums up to 1.
Pascual-Montano A, Carazo JM, Kochi K, Lehmann D and Pascual-marqui RD (2006). "Nonsmooth nonnegative matrix factorization (nsNMF)." _IEEE Trans. Pattern Anal. Mach. Intell_, *28*, pp. 403-415.
# create a completely empty NMFns object
new('NMFns')
## <Object of class:NMFns>
## features: 0
## basis/rank: 0
## samples: 0
## theta: 0.5
# create a NMF object based on random (compatible) matrices
n <- 50; r <- 3; p <- 20
w <- rmatrix(n, r)
h <- rmatrix(r, p)
nmfModel(model='NMFns', W=w, H=h)
## <Object of class:NMFns>
## features: 50
## basis/rank: 3
## samples: 20
## theta: 0.5
# apply Nonsmooth NMF algorithm to a random target matrix
V <- rmatrix(n, p)
## Not run: nmf(V, r, 'ns')
# random nonsmooth NMF model
rnmf(3, 10, 5, model='NMFns', theta=0.3)
## <Object of class:NMFns>
## features: 10
## basis/rank: 3
## samples: 5
## theta: 0.3