DLPC

LP-classifier on dissimilarity (proximity) data

    [W1,W2,W3] = DLPC(D,BIAS,TYPE,PARAM)

Input
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

Output
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.

  DEFAULTS BIAS = 1 TYPE = 'STANDARD' PARAM = []

Description

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.