Canonical Correlation Analysis on Riemannian Manifolds and Its Applications

Canonical correlation analysis (CCA) is a widely used statistical technique to capture correlations between two sets of multi-variate random variables and has found a multitude of applications in computer vision, medical imaging and machine learning. The classical formulation assumes that the data live in a pair of vector spaces which makes its use in certain important scientific domains problematic. For instance, the set of symmetric positive definite matrices (SPD), rotations and probability distributions, all belong to certain curved Riemannian manifolds where vector-space operations are in general not applicable. Analyzing the space of such data via the classical versions of inference models is rather sub-optimal. But perhaps more importantly, since the algorithms do not respect the underlying geometry of the data space, it is hard to provide statistical guarantees (if any) on the results. Using the space of SPD matrices as a concrete example, this paper gives a principled generalization of the well known CCA to the Riemannian setting. Our CCA algorithm operates on the product Riemannian manifold representing SPD matrix-valued fields to identify meaningful statistical relationships on the product Riemannian manifold. As a proof of principle, we present results on an Alzheimer's disease (AD) study where the analysis task involves identifying correlations across diffusion tensor images (DTI) and Cauchy deformation tensor fields derived from T1-weighted magnetic resonance (MR) images.