Norm

The unknow Word

The First Column	The Second Column
leq	less than or equal to
geq	greater than or equal to
Euclidean norm	[ju:klidien]欧几里得

2.5 Norms

Sometimes we need to measure the size of a vector. In machine learning, we usually measure the size of vectors using a function called a norm. Formally, the (L^p) norm is given by

有时我们需要衡量一个向量的大小。在机器学习中，我们经常使用被称为 范数(norm)的函数衡量向量大小。形式上,(L^p)范数定义如下:

[||x||_p=(sum_{i}|x_i|^p)^frac{1}{p} ag{2.30} ]

for (in mathbb{R} , pgeq1).

Norms, including the (L^p) norm, are functions mapping vectors to non-negativevalues. On an intuitive level, the norm of a vector x measures the distance fromthe origin to the point x. More rigorously, a norm is any functionfthat satisfies the following properties:

范数（包括 (L^p) 范数）是将向量映射到非负值的函数。直观上来说，向量 x 的范数衡量从原点到点 x 的距离。更严格地说，范数是满足下列性质的任意函数：

• f(x)=0 => x=0

• (f(x+y)leq f(x)+f(y)) (the triangle inequality 三角不等式)

• (igvee alpha in mathbb{R},f(alpha x)=|alpha|f(x))

The (L^2) norm,with p=2,is known as the Euclidean norm,which is simply the Euclidean distance from the origin to the point identified by x。The (L^2) norm is used so frequently in machine learning that it is often denoted simply as ||x||,with the subscript 2 omitted.It is also common to measure the size of a vector using the squared (L^2) norm,which can be calculated simply as (x^T x).

当 p = 2 时,(L^2)范数被称为 欧几里得范数（ Euclidean norm）。它表示从原点出发到向量 x 确定的点的欧几里得距离。(L^2)范数在机器学习中出现地十分频繁，经常简化表示为 ||x||，略去了下标 2。平方 L2 范数也经常用来衡量向量的大小，可以简单地通过点积(x^T x) 计算。