In a problem I am working on, I have two random variables, X and Y. I need to figure out how closely correlated the two of them are, but they are of different dimensions. The rank of the row space of X is 4350, and the rank of the row space of Y is substantially larger, in the tens of thousands. Both X and Y have the same number of columns.
I need a measure of correlation between the two variables, and Pearson's r requires X and Y to have equal dimension (at least R requires the two r.v.'s to be).
Do I have any hope of doing a correlation between these two, or should I find some way of pruning off observations from Y?