Canonical correlation analysis (CCA) aims to maximize the usual Pearson product-moment correlation (i.e. linear correlation coefficient) of the linear combinations of the two data sets.
Now, consider the fact that this correlation coefficient only measures linear associations - this is the very reason why we also use, for example, Spearman-$\rho$ or Kendall-$\tau$ (rank)correlation coefficients which measure arbitrary monotone (not necessarily linear) connection between variables.
Hence, I was thinking of the following: one limitation of CCA is that it only tries to capture linear association between the formed linear combinations due to its objective function. Wouldn't it be possible to extend CCA in some sense by maximizing, say, Spearman-$\rho$ instead of Pearson-$r$?
Would such procedure lead to anything statistically interpretable and meaningful? (Does it make sense - for example - to perform CCA on ranks...?) I am wondering if it would help when we are dealing with non-normal data...