R: Compute Norms of Row and Column Vectors of a Matrix...

rowNorms {wordspace} | R Documentation

## Compute Norms of Row and Column Vectors of a Matrix (wordspace)

### Description

Efficiently compute the norms of all row or column vectors of a dense or sparse matrix.

### Usage

    rowNorms(M, method = "euclidean", p = 2)

    colNorms(M, method = "euclidean", p = 2)

### Arguments

`M` | a dense or sparse numeric matrix
`method` | norm to be computed (see “Norms” below for details)
`p` | exponent of the `minkowski` p-norm, a numeric value in the range 1 ≤ p ≤ ∞. Values 0 ≤ p < 1 are also permitted as an extension but do not correspond to a proper mathematical norm (see details below).

### Value

A numeric vector containing one norm value for each row or column of `M`.

### Norms

Given a row or column vector x, the following length measures can be computed:

`euclidean`

  The Euclidean norm given by

  |x|_2 = sqrt( SUM(i) (x_i)^2 )

`maximum`

  The maximum norm given by

  |x|_Inf = MAX(i) |x_i|

`manhattan`

  The Manhattan norm given by

  |x|_1 = SUM(i) |x_i|

`minkowski`

  The Minkowski (or L_p) norm given by

  |x|_p = [ SUM(i) |x_i|^p ]^(1/p)

  for p ≥ 1. The Euclidean (p = 2) and Manhattan (p = 1) norms are special cases, and the maximum norm corresponds to the limit for p -> Inf.

  As an extension, values in the range 0 ≤ p < 1 are also allowed and compute the length measure

  |x|_p = SUM(i) |x_i|^p

  For 0 < p < 1 this formula defines a p-norm, which has the property |r * x| = |r|^p * |x| for any scalar factor r instead of being homogeneous. For p = 0, it computes the Hamming length, i.e. the number of nonzero elements in the vector x.

### Author(s)

Stefan Evert (http://purl.org/stefan.evert)

### See Also

`dist.matrix`, `normalize.rows`

### Examples

    rowNorms(DSM_TermContextMatrix, "manhattan")

    # fast and memory-friendly nonzero counts with "Hamming length"
    rowNorms(DSM_TermContextMatrix, "minkowski", p=0)
    colNorms(DSM_TermContextMatrix, "minkowski", p=0)
    sum(colNorms(DSM_TermContextMatrix, "minkowski", p=0)) # = nnzero(DSM_TermContextMatrix)

[Package wordspace version 0.2-6 Index]