Relation between kernel CCA and kernel FDA

In this paper, relation between multi-class linear and kernel Fisher discriminant analysis (FDA) and linear and kernel canonical correlation analysis (CCA) is established. It is shown that in a multi-class classification problem, the CCA between feature vectors (or a nonlinearly mapped version of them) as one-channel and the class label vectors as the second channel is equivalent to multi-class FDA. The multi-class Fisher distance is found to be decomposed into a sum of terms, each of which is determined by a canonical correlation. This result is extended to the kernel formulation without explicit computation of the nonlinear mappings. A simple example is presented to numerically verify the results.