  Sparse Canonical Correlation Analysis (CCA) has received considerable
attention in high-dimensional data analysis to study the relationship between
two sets of random variables. However, there has been remarkably little
theoretical statistical foundation on sparse CCA in high-dimensional settings
despite active methodological and applied research activities. In this paper,
we introduce an elementary sufficient and necessary characterization such that
the solution of CCA is indeed sparse, propose a computationally efficient
procedure, called CAPIT, to estimate the canonical directions, and show that
the procedure is rate-optimal under various assumptions on nuisance parameters.
The procedure is applied to a breast cancer dataset from The Cancer Genome
Atlas project. We identify methylation probes that are associated with genes,
which have been previously characterized as prognosis signatures of the
metastasis of breast cancer.
