NMath Reference Guide

## NMFNonsmooth |

Class NMFNonsmoothUpdate encapulates an NMF update algorithm which
minimizes a cost functional designed to explicitly represent sparseness,
in the form on nonsmoothness, which is controlled by a single parameter.

Inheritance Hierarchy

Syntax

The NMFNonsmoothUpdate type exposes the following members.

Constructors

Name | Description | |
---|---|---|

NMFNonsmoothUpdate | Constructs a NMFNonsmoothUpdate instance with smoothing parameter theta equal to zero. | |

NMFNonsmoothUpdate(Double) | Constructs a NMFNonsmoothUpdate instance with the specified smoothing parameter. |

Properties

Name | Description | |
---|---|---|

SmoothingMatrix |
Gets the smoothing matrix.
The smoothing matrix is a positive
symmetric matrix S defined as:
S = (1 - theta)I + (theta/q)11'where I is the identity matrix, 1 is a vector of ones and the parameter t is between 0 and 1, inclusive. See class description. | |

Theta |
The Nonsmooth Nonnegative Matrix Factorization is defined as
V = WSHwhere V, W, and H are the same as in the original NMF model. The positive symmetric matrix S is a "smoothing" matrix defined as: S = (1 - theta)I + (theta/q)11' |

Methods

Name | Description | |
---|---|---|

Clone | Returns a deep copy of self. | |

Cost | Returns the value of the cost function for the non-negative factorization of V = WH. The cost function is minimized by the factorization. | |

InitialFactors | Provides the initial factors for the iteration process that computes W and H in the non-negative matrix factorization V = WH. | |

IterationStep | Updates the factors W and H in the non-negative matrix factorization V = WH. |

Remarks

The Nonsmooth Nonnegative Matrix Factorization is defined as

V = WSHwhere V, W, and H are the same as in the original NMF model. The positive symmetric matrix S is a "smoothing" matrix defined as:

S = (1 - theta)I + (theta/q)11'where I is the identity matrix, 1 is a vector of ones and the parameter theta is between 0 and 1, inclusive. The interpretation of theta as a smoothing parameter is as follows: If x is a positive, nonzero, vector and y = Sx, then y = x when theta = 0 and there is no smoothing. As theta -> 1, the vector y tends to the constant vector with all elements equal to the average of the elements of x. This is the smoothest possible vector in the sense of nonsparseness because all the entries are equal to the same nonzero value, instead of having values close to zero and others clearly not zero. See the paper Nonsmooth Nonnegative Matrix Factorization (nsNMF) Alberto Pascual-Montano, J.M Carazo, Kieko Kochi, Dietrich Lehmann, and Roberto D. Pascual-Marui IEEE Transactions on Pattern Analysis and Machine Intelligence Vol. 28, No. 3, March 2006.

See Also