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LP-classifier on dissimilarity (proximity) data


 D Dissimilarity (proximity) dataset
 BIAS YES or NO (optional; default: 1 (YES))
 TYPE Type of a classifier
 'SIMPLE' the most simple formulation; no sparse solution; PARAM = [];
 'STANDARD' minimization of the training misclassification errors;  no sparse solution; PARAM = [];
 'C-SPARSE' sparse solution; a formulation similar to the LP_1 SVM;
 PARAM is a tradeoff parameter, similar as in the traditional
 SVM; (optional; DEFAULT: 1).
 'MU-SPARSE' sparse solution; a formulation similar to the LP_1 SVM,  based on the paper of Graepel, Herbrich, Smola etc  'Classification on proximity data with LP-machines'.
 PARAM is a tradeoff parameter, usually PARAM = 0.05 or 0.1.  It is an upper bound on the misclassfied training objects.  So, for well separable problems, PARAM = 0.01 or PARAM = 0.02.  (optional; DEFAULT: the LOO 1-NN error * 1.3).
 PARAM Parameter connected to the TYPE, as above

 W1 LP-Classifier in the complete dissimilarity space
 W2 LP-Classifier in a reduced dissimilarity space
 W3 Object selection prmapping; the indices of support objects are in +W3.


Classification problem on a N x M dissimilarity dataset D with LP-machines.  D should be described by both label and feature lists. If D is a square,  symmetric matrix, then the feature list should be the same as the label list.

Assume a 2-class problem. Let DLPC select J support objects. Then
W1 is an M x 2 classifier in the original dissimilarity space, W2 is an J x 2 classifier in the dissimilarity space defined by the J support objects  and W3 is an M x R feature selection such that W1 = W3 * W2.  Note that the indices of the support objects can be retrieved by +W3.

A linear classifier is built on D

       f(D(x,*)) = diag(Y) * D(x,*) * W + W0,

where Y are labels (+/- 1) and W are the weights. If BIAS is 1, then W0 is also  sought, otherwise it equals 0, hence the hyperplane is forced to go through the origin.

For C-class problems, C classifiers are trained, one against all others.  In such a case, only W1 is returned and W3 in now NOT a feature selection,  but directly the indices of the support objects.

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