Norm

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The unknow Word

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:
\[||x||_p=(\sum_{i}|x_i|^p)^\frac{1}{p}\tag{2.30}\]
for \(\in \mathbb{R} , p\geq1\).

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)

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

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。

Norm

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原文地址：https://www.cnblogs.com/hugeng007/p/9534933.html