  We introduce Block Sparse Canonical Correlation Analysis which estimates
multiple pairs of canonical directions (together a "block") at once, resulting
in significantly improved orthogonality of the sparse directions which, we
demonstrate, translates to more interpretable solutions. Our approach builds on
the sparse CCA method of (Solari, Brown, and Bickel 2019) in that we also
express the bi-convex objective of our block formulation as a concave
minimization problem over an orthogonal k-frame in a unit Euclidean ball, which
in turn, due to concavity of the objective, is shrunk to a Stiefel manifold,
which is optimized via gradient descent algorithm. Our simulations show that
our method outperforms existing sCCA algorithms and implementations in terms of
computational cost and stability, mainly due to the drastic shrinkage of our
search space, and the correlation within and orthogonality between pairs of
estimated canonical covariates. Finally, we apply our method, available as an
R-package called BLOCCS, to multi-omic data on Lung Squamous Cell
Carcinoma(LUSC) obtained via The Cancer Genome Atlas, and demonstrate its
capability in capturing meaningful biological associations relevant to the
hypothesis under study rather than spurious dominant variations.
