Introduction

In the field of mathematics and machine learning, the concept of a norm is a central measure used to express the length or size of a vector. Among various types of norms, p-norms constitute a broad class that further expands the understanding and application of vector norms. This blog post aims to introduce the concept of vector norms and delve into the details of p-norms.

Applications

Understanding Vector Norms

In a simple sense, a norm is a function that assigns a positive length or size to a vector. In a \(n\)-dimensional real or complex space, a norm \(||x||\) of a vector \(x\) is usually denoted as:

Introducing p-norms

The concept of p-norms generalizes the idea of norms by introducing a parameter \(p\). For a vector \(x = [x_1, x_2, \dots, x_n]\) in \(n\)-dimensional space, the p-norm (\(1 \leq p < \infty\)) is defined as:

Applications of p-norms

The different values of \(p\) in p-norms lead to different types of norms. For example, when \(p=1\), it is the Manhattan Norm, and when \(p=2\), it is the familiar Euclidean Norm. These norms are widely used in machine learning algorithms such as regression, classification, and clustering.

References

• Norm - Wikipedia
• Lp space - Wikipedia