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Canonical Correlation Analysis on SPD(n) Manifolds

  • Hyunwoo J. Kim
  • Nagesh Adluru
  • Barbara B. Bendlin
  • Sterling C. Johnson
  • Baba C. Vemuri
  • Vikas Singh

Abstract

Canonical correlation analysis (CCA) is a widely used statistical technique to capture correlations between two sets of multivariate 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 suboptimal. Using the space of SPD matrices as a concrete example, we present 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 correlations. As a proof of principle, we present experimental results on a neuroimaging data set to show the applicability of these ideas.

Keywords

Riemannian Manifold Canonical Correlation Analysis Geodesic Curve Product Manifold Diffusion Tensor Image 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work was supported in part by NIH grants AG040396 (VS), AG037639 (BBB), AG021155 (SCJ), AG027161 (SCJ), NS066340 (BCV), NSF CAREER award 1252725 (VS). Partial support was also provided by UW ADRC (AG033514), UW ICTR, Waisman Core grant (P30 HD003352), and the Center for Predictive Computational Phenotyping (CPCP) at UW-Madison (AI117924). The contents do not represent views of the Dept. of Veterans Affairs or the United States Government.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Hyunwoo J. Kim
    • 1
  • Nagesh Adluru
    • 1
  • Barbara B. Bendlin
    • 1
  • Sterling C. Johnson
    • 1
  • Baba C. Vemuri
    • 2
  • Vikas Singh
    • 1
  1. 1.University of Wisconsin-MadisonMadisonUSA
  2. 2.University of FloridaGainesvilleUSA

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