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 developed too. One such setting includes kernels that are incorporated to account for nonlinear data dependencies. The resultant graph-kernel 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.
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Manuscript received March 27, 2018; revised June 24, 2018; accepted June 25, 2018. Date of publication July 9, 2018; date of current version July 18, 2018. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Olivier Lezoray. This work was partially supported by NSF under Grants 1500713 and 1514056. This paper was presented in part at the IEEE Statistical Signal Processing Workshop, Freiburg, Germany, June 10–13, 2018. (Corresponding author: Gang Wang.) J. Chen, Y. Shen, and G. B. Giannakis are with the Digital Technology Center and the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail:, email@example.com; firstname.lastname@example.org; email@example.com).
- Dimensionality reduction
- Laplacian regularization
- correlation analysis
- generalized eigen-decomposition
- signal processing over graphs