Support vector learning for ordinal regression

This task is referred to as ordinal regression and is complementary to the standard machine learning tasks of classification and metric regression.

In ordinal regression, authors consider a problem which shares properties of both classification  and metric regression.and then present a distribution independent model for ordinal regression, which is based on a loss function that acts on pairs of ranks.


A Risk Formulation for Ordinal Regression

Thus, the problem of ordinal regression can be reduced to a classification problem on pairs of

objects.





Support Vector Machines for Ordinal Regression

1.Consider a linear function

                      

2.  incurs no error if

         

3. Finite margin 
between the n-dimensional feature vectors Xi(1)-Xi(2)of classes Zi=+1and Zi=-1

4. define parallel hyperplanes passing through each pair  (Xi(1), Xi(2)) by

5.minimizing the squared norm

 under constraint 4.

This paper showed that every ordinal regression problem corresponds to a unique preference learning problem on pairs of objects.
This result builds the link between ordinal regression and classification methods on pairs of objects and allows for a theoretical treatment in the framework of classification.