sparseness: Sparseness

In NMF: Algorithms and Framework for Nonnegative Matrix Factorization (NMF)

Sparseness

Description

Generic function that computes the sparseness of an object, as defined by Hoyer (2004). The sparseness quantifies how much energy of a vector is packed into only few components.

Usage

  sparseness(x, ...)

Arguments

x	an object whose sparseness is computed.
...	extra arguments to allow extension

Details

In Hoyer (2004), the sparseness is defined for a real vector x as:

Sparseness(x) = \frac{\sqrt{n} - \frac{\sum |x_i|}{\sqrt{\sum x_i^2}}}{\sqrt{n}-1}

, where n is the length of x.

The sparseness is a real number in [0,1]. It is equal to 1 if and only if x contains a single nonzero component, and is equal to 0 if and only if all components of x are equal. It interpolates smoothly between these two extreme values. The closer to 1 is the sparseness the sparser is the vector.

The basic definition is for a numeric vector, and is extended for matrices as the mean sparseness of its column vectors.

Value

usually a single numeric value – in [0,1], or a numeric vector. See each method for more details.

Methods

sparseness

signature(x = "numeric"): Base method that computes the sparseness of a numeric vector.

It returns a single numeric value, computed following the definition given in section Description.

sparseness

signature(x = "matrix"): Computes the sparseness of a matrix as the mean sparseness of its column vectors. It returns a single numeric value.

sparseness

signature(x = "NMF"): Compute the sparseness of an object of class NMF, as the sparseness of the basis and coefficient matrices computed separately.

It returns the two values in a numeric vector with names ‘basis’ and ‘coef’.

References

Hoyer P (2004). "Non-negative matrix factorization with sparseness constraints." _The Journal of Machine Learning Research_, *5*, pp. 1457-1469. <URL: http://portal.acm.org/citation.cfm?id=1044709>.

See Also

Other assess: entropy, purity

NMF documentation built on Sept. 11, 2024, 8:34 p.m.