# Norm and Li-norm

| Tags Machine Learning

In mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space, other than the zero vector. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors(in addition to the zero vector). Norm may come in many forms and many names, including these popular names: Euclidean distance, Mean-squared Error, etc. Usually, norm appears in this form:

$$| x |$$

where $$x$$$is a vector or matrix, for example, a Euclidean norm of a vector $$a = [1, 2, 3]^T$$$ is:

$$| a |_2 = \sqrt{1^2 + 2^2 + 3^2}$$

## L0-norm

$$| x |_0 = \sum x_i^0$$

That is the number of non-zero elements.

## L1-norm

$$| x |_1 = \sum |x_i|$$

That is the sum of each element's absolute value.

## L2-norm

$$| x |_2 = \sqrt{\sum x_i^2}$$

It is also called Euclidean norm.

## Linfinity-norm

For $$p \geq 1$$$, $$| x |_p = (\sum |x_i|^p)^{\frac{1}{p}}$$ For infinity, the definition looks very tricky, $$|x|_{\infty} = \sqrt[\infty]{\sum |x_i|^{\infty}}$$ Fortunately, this equation can be transformed into a very simple format, let's suppose $$m$$$ be the largest number of $$x$$$, by the property of infinity we have $$m^{\infty} \gg x_i^{\infty}$$$, when $$x_i \neq m$$$.So, $$| x |_{\infty} = \sqrt[\infty]{k|m|^{\infty}}$$$, where k is the number of largest numbers in $$x$$$, still thanks to the property of infinity, we have $$\sqrt[\infty]{k} = 1$$$, at last, the Linfinity-norm is:

$$|x|_{\infty} = max(|x_i|)$$

To Be Continued