  Canonical correlation analysis (CCA for short) describes the relationship
between two sets of variables by finding some linear combinations of these
variables that maximizing the correlation coefficient. However, in
high-dimensional settings where the number of variables exceeds sample size, or
in the case of that the variables are highly correlated, the traditional CCA is
no longer appropriate. In this paper, an adaptive sparse version of CCA (ASCCA
for short) is proposed by using the trace Lasso regularization. The proposed
ASCCA reduces the instability of the estimator when the covariates are highly
correlated, and thus improves its interpretation. The ASCCA is further
reformulated to an optimization problem on Riemannian manifolds, and an
manifold inexact augmented Lagrangian method is then proposed for the resulting
optimization problem. The performance of the ASCCA is compared with the other
sparse CCA techniques in different simulation settings, which illustrates that
the ASCCA is feasible and efficient.
