L1 norm: Difference between revisions

Latest revision as of 18:43, 11 May 2014

Definition

Suppose $n$ is a positive integer. The $L^{1}$ -norm, denoted $| \cdot |_{1}$ , is a function from $R^{n}$ to $R$ defined as:

$| x |_{1} = \sum_{i = 1}^{n} | x_{i} |$

Definition in terms of the signum vector function

The $L^{1}$ -norm $| x |_{1}$ can be defined as the dot product $\overset{g}{s} (\vec{x}) \cdot \vec{x}$ where $\overset{g}{s}$ denotes the signum vector function.

Definition as a piecewise linear function

The $L^{1}$ -norm can be defined as a piecewise linear function. The number of pieces of the domain involved is $2^{n}$ interior regions, and an additional $3^{n} - 2^{n}$ boundary regions and boundary intersections (these can be included in the piece definitions for any of the bordering interior regions).

@@ Line 7: / Line 7: @@
 ===Definition in terms of the signum vector function===
-The <math>L^1</math>-norm <math>\| x \|_1</math> can be defined as the [[dot product]] <math>\vec{\operatorname{sgn}}(\vec{x}) \cdot \vec{x}</math> where <math>\vec{sgn}</math> denotes the [[signum vector function]].
+The <math>L^1</math>-norm <math>\| x \|_1</math> can be defined as the [[dot product]] <math>\overline{\operatorname{sgn}}(\vec{x}) \cdot \vec{x}</math> where <math>\overline{\operatorname{sgn}}</math> denotes the [[signum vector function]].
 ===Definition as a piecewise linear function===
-The <math>L^1</math>-norm can be defined as a piecewise linear function. The number of linear pieces involved is <math>2^n</math>.
+The <math>L^1</math>-norm can be defined as a piecewise linear function. The number of pieces of the domain involved is <math>2^n</math> interior regions, and an additional <math>3^n - 2^n</math> boundary regions and boundary intersections (these can be included in the piece definitions for any of the bordering interior regions).