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Mapping overload

Mappings define transformations of vector spaces of possibly different dimensionality. Their size [size_in,size_out] corresponds to these dimensionalities. Consequently they behave somewhat similar as matrices of this size. Linear (affine) mappings are almost identical to such matrices but includes a shift operation. In addition they may carry several other types of information as explained in the mapping definition.

Operations and operators

A number of operations defined for 2-dimensional matrices can be performed on mappings as well. In the below table some examples are given. A, B and R are datasets, W and V are mappings, possibly non-linear. Most of these operations are generalizations of the corresponding operations between matrices. Some of the examples apply for affine mappings only.

> Examples of standard Matlab operations applied to mappings.
R = A*W Generalization of the matrix multiplication.
R = A*[W V] R = [A*W A*V], more on combining mappings.
R = [A B]*[W;V] R = [A*W;B*V], more on combining mappings.
U = W(:,L) Restrict the output space to the dimensions in the index vector L.
U = W+V Add mappings such that A*U = A*W + A*V.
U = 3*W - V/2 Weight and subtract mappings such that A*U = 3*A*W - A*V/2.
U = W*V Execute mappings sequentially, A*U = (A*W)*V in case W and V are fixed or trained mappings. Different rules apply for other mapping types.

The following Matlab operators are defined for mappings:

    +, -, *, .* , /

The mrdivision operator +/+ is defined for scalar division only. Matlab commands that are overloaded for mappings are: double, isempty, size.


The mapping fields might be accessed by structural indexing. Indexing by V = W(I,J) only applies for I = :, so use V = W(:,J). It means that the output space of the mapping is reduced to dimensions listed in J.

Overloaded commands for affine mappings

As the linear (affine) mappings have a very similar meaning as regular Matlab matrices, operations defined for matrices work as similar as possible for such mappings. Consequently operations like U = W*V result also internally in a new affine mapping U if W and V are affine. Transpose (W') and orthogonalization (orth(W)) are thereby also defined for affine mappings.

R.P.W. Duin, January 28, 2013

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