What is a norm? What is l1, l2 and l infinity norm
Answers
l1-norm
Following the definition of norm, -norm of is defined as
This norm is quite common among the norm family. It has many name and many forms among various fields, namely Manhattan norm is it’s nickname. If the -norm is computed for a difference between two vectors or matrices, that is
it is called Sum of Absolute Difference (SAD) among computer vision scientists.
In more general case of signal difference measurement, it may be scaled to a unit vector by:
where is a size of .
which is known as Mean-Absolute Error (MAE).
l2-norm
The most popular of all norm is the -norm. It is used in almost every field of engineering and science as a whole. Following the basic definition, -norm is defined as
-norm is well known as a Euclidean norm, which is used as a standard quantity for measuring a vector difference. As in -norm, if the Euclidean norm is computed for a vector difference, it is known as a Euclidean distance:
or in its squared form, known as a Sum of Squared Difference (SSD) among Computer Vision scientists:
It’s most well known application in the signal processing field is the Mean-Squared Error (MSE) measurement, which is used to compute a similarity, a quality, or a correlation between two signals. MSE is
As previously discussed in -optimisation section, because of many issues from both a computational view and a mathematical view, many -optimisation problems relax themselves to become – and -optimisation instead. Because of this, we will now discuss about the optimisation of .
