R: NMF Model - Nonsmooth Nonnegative Matrix Factorization

NMFns-class {NMF} | R Documentation

## 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.

### Details

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

V &#126; 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≤q θ ≤q 1 in the following way:

  S = (1-θ)I + \frac{θ}{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 θ = 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.

### 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.

### 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

    # 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)
    ## Not run: nmf(V, r, 'ns')

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

[Package NMF version 0.20.6 Index]