Non-Linear Canonical Correlation Analysis Using Alpha-Beta Divergence

We propose a generalized method of the canonical correlation analysis using Alpha-Beta divergence, called AB-canonical analysis (ABCA). From observations of two random variables, x is an element of R-P and y is an element of R-Q, ABCA finds directions, w(x) is an element of R-P and w(y) is an element of R-Q, such that the AB-divergence between the joint distribution of (w(x)(T)x, w(y)(T)y) and the product of their marginal distributions is maximized. The number of significant non-zero canonical coefficients are determined by using a sequential permutation test. The advantage of our method over the standard canonical correlation analysis (CCA) is that it can reconstruct the hidden non-linear relationship between w(x)(T)x and w(y)(T)y, and it is robust against outliers. We extend ABCA when data are observed in terms of tensors. We further generalize this method by imposing sparseness constraints. Extensive simulation study is performed to justify our approach.