Linear correlation. Actually, it seems to be a bit more general than just uncorrelated, i.e. if the input is a m by n matrix X and ground truth a k by n matrix Y, the author requires that XX^T and XY^T to be full-rank. A whitening transformation would yield the identity matrix for XX^T, but that's a bit stronger than what's strictly necessary. My interpretation of XY^T being full-rank meaning X and Y being uncorrelated might indeed be mistaken.

It is mistaken, if X=Y=I then XY' is full rank but each column in X is a linear function of its counterpart in Y.

Could you clarify what is the mathematical definition of "X and Y are uncorrelated" for two matrices?

I am not quite sure there is such a thing. :p I was playing a bit loose with the mathematics here, and trying to find some more intuitive way to explain "XY^T is full-rank", but it got confusing. Sorry about that. I will edit my initial post accordingly. (Ah, can't edit it seems, oh well)