Lp norms

They are ways of measuring the size or length of vectors.

In Regularisation, the number following L (e.g. L1 or L2) refers to the power used in calculating the norm.

$L^1$ norm

"Manhattan norm" or "Taxicab norm"

• Calculated as $||x|{|}_1=\sum |x_i|$
• Sum of the absolute values
• Called Manhattan norm because it's akin to measuring distance following city blocks

$L^2$ norm

"Euclidean norm"

• Calculated as $||x|{|}_2=\sqrt{(\sum x_i^2)}$
• Square root of sum of squared values
• Called Euclidean norm because it's the standard way we measure straight line distance

$L^{\mathrm{\infty }}$ norm

Max norm

• Calculated as $||x|{|}_{\mathrm{\infty }}=\underset{i,j}{max}(|\widetilde{x_{i,j}}-x_{i,j})$
• Maximum difference between any $\tilde{x}$ value and its corresponding to $x$