inner product of multisets
A multiset consists of a set and a function , where is a universal set and if and only if .
We can add two multisets and to get
Note that we can write
We can also define an inner product of multisets via
Note that when and are simply sets, and are the characteristic functions and
where denotes the cardinality of the set.
Using this inner product, we can define the angle between multisets as
In particular, when we have
and when we have
When and are simply sets, the angle between them is given by
With this notion of addition, the collection of multisets in becomes the -module (that is abelian monoid) ; this inner product makes it an inner product space analogous to the Banach space .
The inner product of multisets is closely related to the “bag of words” kernel in machine learning (see n-Cafe).
Revised on October 28, 2009 17:39:46
by Toby Bartels