  Canonical correlation analysis (CCA) is a fundamental statistical tool for
exploring the correlation structure between two sets of random variables. In
this paper, motivated by recent success of applying CCA to learn low
dimensional representations of high dimensional objects, we propose to quantify
the estimation loss of CCA by the excess prediction loss defined through a
prediction-after-dimension-reduction framework. Such framework suggests viewing
CCA estimation as estimating the subspaces spanned by the canonical variates.
Interestedly, the proposed error metrics derived from the excess prediction
loss turn out to be closely related to the principal angles between the
subspaces spanned by the population and sample canonical variates respectively.
  We characterize the non-asymptotic minimax rates under the proposed metrics,
especially the dependency of the minimax rates on the key quantities including
the dimensions, the condition number of the covariance matrices, the canonical
correlations and the eigen-gap, with minimal assumptions on the joint
covariance matrix. To the best of our knowledge, this is the first finite
sample result that captures the effect of the canonical correlations on the
minimax rates.
