L1-regularized quadratic function of multiple variables: Difference between revisions

Revision as of 19:09, 11 May 2014

Definition

A $L^{1}$ -regularized quadratic function of the variables $x_{1},x_{2},\dots ,x_{n}$ is a function of the form:

$f(x_{1},x_{2},\dots ,x_{n}):=\left(\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}x_{i}x_{j}\right)+\left(\sum _{i=1}^{n}b_{i}x_{i}\right)+\lambda \sum _{i=1}^{n}|x_{i}|+c$

In vector form, if we denote by ${\vec {x}}$ the column vector with coordinates $x_{1},x_{2},\dots ,x_{n}$ , then we can write the function as:

${\vec {x}}^{T}A{\vec {x}}+{\vec {b}}^{T}{\vec {x}}+\lambda \|{\vec {x}}\|_{1}+c$

where $A$ is the $n\times n$ matrix with entries $a_{ij}$ and ${\vec {b}}$ is the column vector with entries $b_{i}$ .

Key data

Item	Value
default domain	the whole of $\mathbb {R} ^{n}$

Differentiation

Partial derivatives and gradient vector

The partial derivative with respect to the variable $x_{i}$ , and therefore also the $i^{th}$ coordinate of the gradient vector (if it exists), is given as follows when $x_{i}\neq 0$ :

${\frac {\partial f}{\partial x_{i}}}=\left(\sum _{j=1}^{n}a_{ij}x_{j}\right)+b_{i}+\operatorname {sgn} (x_{i})$

The partial derivative is undefined when $x_{i}=0$ .

The gradient vector exists if and only if all the coordinates are nonzero.

@@ Line 1: / Line 1: @@
 ==Definition==
-A <math>L^1</math>-'''regularized quadratic functions''' of the variables <math>x_1,x_2,\dots,x_n</math> is a function of the form:
+A <math>L^1</math>-'''regularized quadratic function''' of the variables <math>x_1,x_2,\dots,x_n</math> is a function of the form:
-<math>\left(\sum_{i=1}^n \sum_{j=1}^n a_{ij} x_ix_j\right) + \left(\sum_{i=1}^n b_ix_i\right) + \lambda \sum_{i=1}^n |x_i| + c</math>
+<math>f(x_1,x_2,\dots,x_n) := \left(\sum_{i=1}^n \sum_{j=1}^n a_{ij} x_ix_j\right) + \left(\sum_{i=1}^n b_ix_i\right) + \lambda \sum_{i=1}^n |x_i| + c</math>
 In vector form, if we denote by <math>\vec{x}</math> the column vector with coordinates <math>x_1,x_2,\dots,x_n</math>, then we can write the function as:
@@ Line 10: / Line 10: @@
 where <math>A</math> is the <math>n \times n</math> matrix with entries <math>a_{ij}</math> and <math>\vec{b}</math> is the column vector with entries <math>b_i</math>.
+==Key data==
+{| class="sortable" border="1"
+! Item !! Value
+|-
+| default [[domain]] || the whole of <math>\R^n</math>
+|}
+==Differentiation==
+===Partial derivatives and gradient vector===
+The partial derivative with respect to the variable <math>x_i</math>, and therefore also the <math>i^{th}</math> coordinate of the [[gradient vector]] (if it exists), is given as follows when <math>x_i \ne 0</math>:
+<math>\frac{\partial f}{\partial x_i} = \left(\sum_{j=1}^n a_{ij}x_j\right) + b_i + \operatorname{sgn}(x_i)</math>
+The partial derivative is undefined when <math>x_i = 0</math>.
+The gradient vector exists if and only if ''all the coordinates are nonzero''.