NMF (version 0.28)

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.

Methods

fitted

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.

show

signature(object = "NMFns"): Show method for objects of class NMFns

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)
if (FALSE) nmf(V, r, 'ns')

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