  The classical Canonical Correlation Analysis (CCA) identifies the
correlations between two sets of multivariate variables based on their
covariance, which has been widely applied in diverse fields such as computer
vision, natural language processing, and speech analysis. Despite its
popularity, CCA can encounter challenges in explaining correlations between two
variable sets within high-dimensional data contexts. Thus, this paper studies
Sparse Canonical Correlation Analysis (SCCA) that enhances the interpretability
of CCA. We first show that SCCA generalizes three well-known sparse
optimization problems, sparse PCA, sparse SVD, and sparse regression, which are
all classified as NP-hard problems. This result motivates us to develop strong
formulations and efficient algorithms. Our main contributions include (i) the
introduction of a combinatorial formulation that captures the essence of SCCA
and allows the development of approximation algorithms; (ii) the derivation of
an equivalent mixed-integer semidefinite programming model that facilitates a
specialized branch-and-cut algorithm with analytical cuts; and (iii) the
establishment of the complexity results for two low-rank special cases of SCCA.
The effectiveness of our proposed formulations and algorithms is validated
through numerical experiments.
