Next: SVD within tensor algebra
Up: tr00dl2
Previous: Shortcomings of current statistical
The aim of this section is to explain the basics of the SVD-kmodes method as an extension of
SVD. The presentation can be limited to and as the case is then a
straightforward extension. Further details are in [16]. First of all the development
of this generalisation of the SVD (Singular Value Decomposition) is described within tensor
algebra framework in finite dimension. It enables us to extend matrix algebra calculus in an easy
way. A tensor of order one is a vector, a tensor of order two is a matrix, a tensor of order three
is three-way array etc...
Let
,
, and
be the canonical bases respectively
of , and ; with and let us define the
bilinear map by:
|
(1) |
(where is the inner product in ) ; consider now the canonical bilinear maps built
with the and , they constitute a base of a space noted the tensor product
of the spaces and . Without going further into algebraic concepts, notice that because of
symmetry in equation (1) can be considered as a linear map onto , so
that one has the universal property of the tensor product: transforming a bilinear map
(multilinear in general) into a linear map.
An matrix of elements
, can be written algebraically,
|
(2) |
and is
said to belong to the space , tensorial product of the spaces and
1. Notice that the array, the linear map
associated, the tensor are noted because of isomorphisms. In the same manner a three-way array
of elements
, can be written algebraically,
|
(3) |
The vectors of the space
with the form
(where
) are called decomposed tensors, and are said to be of rank one - a sum of linearly
independent decomposed tensors would give a rank tensor-2. To finish
with basic tools of tensor algebra, let us also introduce the generalisation of a product of
a vector by a matrix: the product of vector (or a tensor) by a tensor, also called contraction
and noted ``..". For example let
, then
with:
Arithmetically this can be seen considering as a matrix with rows and columns, then
calculating the image of by this matrix gives a representation of . Note that
is the inner product between the tensors
.
Subsections
Next: SVD within tensor algebra
Up: tr00dl2
Previous: Shortcomings of current statistical
Didier Leibovici
2001-09-04