  Canonical correlation analysis (CCA) is a powerful technique for discovering
whether or not hidden sources are commonly present in two (or more) datasets.
Its well-appreciated merits include dimensionality reduction, clustering,
classification, feature selection, and data fusion. The standard CCA however,
does not exploit the geometry of the common sources, which may be available
from the given data or can be deduced from (cross-) correlations. In this
paper, this extra information provided by the common sources generating the
data is encoded in a graph, and is invoked as a graph regularizer. This leads
to a novel graph-regularized CCA approach, that is termed graph (g) CCA. The
novel gCCA accounts for the graph-induced knowledge of common sources, while
minimizing the distance between the wanted canonical variables. Tailored for
diverse practical settings where the number of data is smaller than the data
vector dimensions, the dual formulation of gCCA is also developed. One such
setting includes kernels that are incorporated to account for nonlinear data
dependencies. The resultant graph-kernel (gk) CCA is also obtained in closed
form. Finally, corroborating image classification tests over several real
datasets are presented to showcase the merits of the novel linear, dual, and
kernel approaches relative to competing alternatives.
