PE_KERNELM

Pseudo-Euclidean kernel mapping

     K = PE_KERNELM(A,B)
     W = B*PE_KERNELM
     W = PE_KERNELM([],B)
     K = A*W

Input
A	Pseudo-Euclidean dataset of size NxK
B	Pseudo-Euclidean dataset of size MxK

Output
W	PE mapping
K	Kernel matrix, size [N M]

Description

Computation of a kernel matrix in a pseudo-Euclidean space. The signature  of this space should be stored in the datasets A and B, see SETSIG.  K = A*J*B', where J is a diagonal matrix with 1's, followed by -1's.

J	= diag ([ONES(SIG(1),1); -ONES(SIG(2),1)]);
The	two-element vector SIG stores the signature of the space. This is the
number	of 'positive' dimensions, followed by the number of 'negative'
dimensions.	It is computed by a pseudo-Eucledean embedding, e.g. PSEM,
and	stored in the related mapping and datasets that are projected in this
space.

The resulting kernel matrix K is indefinite in case A == B. This routine  may be used in support vector routines and other kernelized procedures.  Note that most of such routines are not optimal for indefinite kernels.

Example(s)

 a = gendatb;                  % generate dataset
 d = a*proxm(a,'m',1);         % compute L1 distance matrix
 w = psem(d);                  % embed in PE space
 b = d*w;                      % project data in this space
 [trainset testset] = gendat(b,0.5);     % split in trainset and testset
 ktrain = pe_kernelm(trainset,trainset); % compute train kernel
 w = svc(ktrain,0);            % compute SV classifier
 ktest = pe_kernelm(testset,trainset);   % compute test kernel
 ktest*w*testc                 % inspect error of testset   

See also

datasets, mappings, pe_em, pe_distm,