rowNorms {wordspace}

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 \le p \le \infty.

Values 0 \le 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\|_{\infty} = \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 = \left( \sum_i |x_i|^p \right)^{1/p}

for p \ge 1. The Euclidean (p = 2) and Manhattan (p = 1) norms are special cases, and the maximum norm corresponds to the limit for p \to \infty.

As an extension, values in the range 0 \le 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\cdot x\| = |r|^p \cdot \|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)

Stephanie Evert (https://purl.org/stephanie.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)