  A new approach to the sparse Canonical Correlation Analysis (sCCA)is proposed
with the aim of discovering interpretable associations in very high-dimensional
multi-view, i.e.observations of multiple sets of variables on the same
subjects, problems. Inspired by the sparse PCA approach of Journee et al.
(2010), we also show that the sparse CCA formulation, while non-convex, is
equivalent to a maximization program of a convex objective over a compact set
for which we propose a first-order gradient method. This result helps us reduce
the search space drastically to the boundaries of the set. Consequently, we
propose a two-step algorithm, where we first infer the sparsity pattern of the
canonical directions using our fast algorithm, then we shrink each view, i.e.
observations of a set of covariates, to contain observations on the sets of
covariates selected in the previous step, and compute their canonical
directions via any CCA algorithm. We also introduceDirected Sparse CCA, which
is able to find associations which are aligned with a specified experiment
design, andMulti-View sCCA which is used to discover associations between
multiple sets of covariates. Our simulations establish the superior convergence
properties and computational efficiency of our algorithm as well as accuracy in
terms of the canonical correlation and its ability to recover the supports of
the canonical directions. We study the associations between metabolomics,
trasncriptomics and microbiomics in a multi-omic study usingMuLe, which is an
R-package that implements our approach, in order to form hypotheses on
mechanisms of adaptations of Drosophila Melanogaster to high doses of
environmental toxicants, specifically Atrazine, which is a commonly used
chemical fertilizer.
