rowNorms: 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

a dense or sparse numeric matrix

method

norm to be computed (see “Norms” below for details)

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 $0\le p < 1$ compute the length measure $$ \|x\|_p = \sum_i |x_i|^p $$ This formula does not define a proper mathematical norm because it is not homogeneous ($\|r\cdot x\| \ne |r|\cdot \|x\|$ for a scalar factor $r$). However, it does satisfy the triangle inequality and is thus still a sensible measure of vector length. In the special case $p = 0$, the length $\|x\|_0$ corresponds to the number of nonzero elements in the vector $x$.

Examples

Run this code

# NOT RUN {
rowNorms(DSM_TermContextMatrix, "manhattan")

colNorms(DSM_TermContextMatrix, "minkowski", p=0) # number of nonzero cells

# }

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