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I have recently been looking into canonical correlation analysis (CCA) as a way to map between different spaces.

As I understand it, CCA maps data from both distinct spaces to a common (possibly lower dimensional) space where they can be compared. It works in a similar way to PCA, choosing the direction from each input space which maximises the correlation between datasets, subject to the chosen directions being uncorrelated.

Now, the descriptions I've seen suggest that CCA can learn any linear transformation. However, I can't see how it's possible for this algorithm to learn shears and rotations in two dimensions. They don't seem to fit the paradigm of a sum of independent, one-dimensional regressions.

So my question is: is CCA able to model arbitrary shears and rotations in 2D space?

Also, if you could outline why your answer is true that would be great.

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I think I understand my mistake. In Hotelling's paper he says that:

"Every invariant under general linear internal transformations... will be seen to be a function of the canonical correlations."

So CCA is not trying to model arbitrary linear transformations at all. It tries to project both datasets into a new space in a way that conserves their invariant quantities (i.e. those not affected by linear transformations).

In short, CCA will do its best to ignore linear transformations of each dataset - just the opposite of how I thought it behaved.

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