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

 sparseness R Documentation

## 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:

(srqt(n) - ||x||_1 / ||x||_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>.

Other assess: `entropy`, `purity`