Vector Norms

$L_1$ Norm (Taxicab Norm or Manhattan norm)	$L_2$ Norm (Euclidean norm)	Max Norm
The $L_1$ norm is calculated as the sum of the absolute vector values.	The $L_2$ norm is calculated as the square root of the sum of the squared vector values.	The max norm is calculated as returning the maximum value of the vector, hence the name.
$\begin{Vmatrix} \boldsymbol{v} \end{Vmatrix}_1 = \begin{vmatrix} a_1 \end{vmatrix} + \begin{vmatrix} a_2 \end{vmatrix} + \begin{vmatrix} a_3 \end{vmatrix}$	$\begin{Vmatrix} \boldsymbol{v} \end{Vmatrix}_2 = \sqrt{\sum\limits_{i=1}^3{a_i^2}} = \sqrt{a_1^2 + a_2^2 + a_3^2}$	$\begin{Vmatrix} \boldsymbol{v} \end{Vmatrix}_{\infty} = \max(a_1, a_2, a_3)$
In effect, the $L_1$ norm is a calculation of the Manhattan distance from the origin of the vector space.	The L2 norm is also known as the Euclidean norm as it is calculated as the Euclidean distance from the origin.
The $L_1$ norm is often used when fitting machine learning algorithms as a regularization method, e.g. a method to keep the coefficients of the model small, and in turn, the model less complex.	Like the $L_1$ norm, the $L_2$ norm is often used when fitting machine learning algorithms as a regularization method, e.g. a method to keep the coefficients of the model small and, in turn, the model less complex.	Max norm is also used as a regularization in machine learning, such as on neural network weights, called max norm regularization.
	By far, the $L_2$ norm is more commonly used than other vector norms in machine learning.

$L_1$ norm

• The $L_1$ norm of a vector can be calculated in NumPy using the norm() function with a parameter to specify the norm order, in this case 1.

# l1 norm of a vector
from numpy import array
from numpy.linalg import norm
a = array([1, 2, 3])
print(a) # [1 2 3]
l1 = norm(a, 1)
print(l1) # 6

$L_2$ Norm

• The $L_2$ norm of a vector can be calculated in NumPy using the norm() function with default parameters.

# l2 norm of a vector
from numpy import array
from numpy.linalg import norm
a = array([1, 2, 3])
print(a) # [1 2 3]
l2 = norm(a)
print(l2) # 3.7416573867739413

Max Norm

The max norm of a vector can be calculated in NumPy using the norm() function with the order parameter set to inf.

# max norm of a vector
from numpy import inf
from numpy import array
from numpy.linalg import norm
a = array([1, 2, 3])
print(a) # [1 2 3]
maxnorm = norm(a, inf)
print(maxnorm) # 3.0