NMF (version 0.20.6)

NMFns-class: NMF Model - Nonsmooth Nonnegative Matrix Factorization

Description

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.

Creating objects from the Class

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, 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.

Algorithm

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.

Details

The Nonsmooth NMF algorithm is a modification of the standard divergence based NMF algorithm (see NMF). Given a non-negative $n \times p$ matrix $V$ and a factorization rank $r$, it fits the following model:

$$V \equiv W S(\theta) H,$$ where:

• $W$and$H$are such as in the standard model, i.e. non-negative matrices of dimension$n \times r$and$r \times p$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 $\theta \to 1$, 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.

References

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.

See Also

Other NMF-model: initialize,NMFOffset-method, NMFOffset-class, NMFstd-class

Examples

# roxygen generated flag
options(R_CHECK_RUNNING_EXAMPLES_=TRUE)

# create a completely empty NMFns object
new('NMFns')

# 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)

# apply Nonsmooth NMF algorithm to a random target matrix
V <- rmatrix(n, p)
nmf(V, r, 'ns')

# random nonsmooth NMF model
rnmf(3, 10, 5, model='NMFns', theta=0.3)