  Multi-block CCA constructs linear relationships explaining coherent
variations across multiple blocks of data. We view the multi-block CCA problem
as finding leading generalized eigenvectors and propose to solve it via a
proximal gradient descent algorithm with $\ell_1$ constraint for high
dimensional data. In particular, we use a decaying sequence of constraints over
proximal iterations, and show that the resulting estimate is rate-optimal under
suitable assumptions. Although several previous works have demonstrated such
optimality for the $\ell_0$ constrained problem using iterative approaches, the
same level of theoretical understanding for the $\ell_1$ constrained
formulation is still lacking. We also describe an easy-to-implement deflation
procedure to estimate multiple eigenvectors sequentially. We compare our
proposals to several existing methods whose implementations are available on R
CRAN, and the proposed methods show competitive performances in both
simulations and a real data example.
