A norm bijection is the composition of a norm and an increasing bijection.
Let be a norm and an increasing bijection. A norm bijection is a function defined by .
A restricted norm bijection is any function , which can expressed in the form with the functions and defined below (redundant conditions in parentheses).
, such that:
( is invertible)
, such that
In particular, is a norm and is the increasing bijection in the definition of norm bijection.
The motivation for the definition of a norm bijection is efficiency. The efficiency follows from the observation that the norm value is rarely of interest itself. Rather, it is the ordering of norms that is of interest. In particular, the p-norms (also called Minkowski norms) compute a :th root for each evaluation, which is a rather slow operation. However, if all you have is an inequality , then this is equivalent to and you can avoid the root computation.
A norm bijection allows you, in particular, to move between the norm bijection value and the norm value. However, we have noticed that to really produce efficient code with norm bijections, it is essential to restrict them further. Therefore, when talking about norm bijections, in Pastel this always means a restricted norm bijection as defined above. Pastel implements the following restricted norm bijections:
A norm bijection of the Euclidean norm
A norm bijection of the Manhattan norm
A norm bijection of the maximum norm
A norm bijection of the Minkowski norm (p-norm)