Riemannian Computing in Computer Vision pp 69-100 | Cite as

# Canonical Correlation Analysis on SPD(*n*) Manifolds

## 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## 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.

## References

- 1.Afsari B (2011) Riemannian
*l*_{p}center of mass: existence, uniqueness, and convexity. Proc Am Math Soc 139(2):655–673zbMATHMathSciNetCrossRefGoogle Scholar - 2.Akaho A (2001) A kernel method for canonical correlation analysis. In: International meeting on psychometric societyGoogle Scholar
- 3.Andrew G, Arora R, Bilmes J, Livescu K (2013) Deep canonical correlation analysis. In: ICMLGoogle Scholar
- 4.Avants BB, Gee JC (2004) Geodesic estimation for large deformation anatomical shape and intensity averaging. NeuroImage 23:S139–150CrossRefGoogle Scholar
- 5.Avants BB, Epstein CL, Grossman M, Gee JC (2008) Symmetric diffeomorphic image registration with cross-correlation: Evaluating automated labeling of elderly and neurodegenerative brain. Med Image Anal 12:26–41CrossRefGoogle Scholar
- 6.Avants BB, Cook PA, Ungar L, Gee JC, Grossman M (2010) Dementia induces correlated reductions in white matter integrity and cortical thickness: a multivariate neuroimaging study with SCCA. NeuroImage 50(3):1004–1016CrossRefGoogle Scholar
- 7.Avants BB, Libon DJ, Rascovsky K, Boller A, McMillan CT, Massimo L, Coslett H, Chatterjee A, Gross RG, Grossman M (2014) Sparse canonical correlation analysis relates network-level atrophy to multivariate cognitive measures in a neurodegenerative population. NeuroImage 84(1):698–711CrossRefGoogle Scholar
- 8.Bach FR, Jordan MI (2003) Kernel independent component analysis. J Mach Learn Res 3:1–48zbMATHMathSciNetGoogle Scholar
- 9.Basser PJ, Mattiello J, LeBihan D (1994) MR diffusion tensor spectroscopy and imaging. Biophys J 66(1):259CrossRefGoogle Scholar
- 10.Bhatia R (2009) Positive definite matrices. Princeton University Press, PrincetonCrossRefGoogle Scholar
- 11.Callaghan PT (1991) Principles of nuclear magnetic resonance microscopy. Oxford University Press, OxfordGoogle Scholar
- 12.Chung MK, Worsley KJ, Paus T, Cherif C, Collins DL, Giedd JN, Rapoport JL, Evans AC (2001) A unified statistical approach to deformation-based morphometry. NeuroImage 14(3):595–606CrossRefGoogle Scholar
- 13.Cook PA, Bai Y, Nedjati-Gilani S, Seunarine KK, Hall MG, Parker GJ, Alexander DC (2006) Camino: open-source diffusion-MRI reconstruction and processing. In: ISMRM, pp 2759Google Scholar
- 14.Do Carmo MP (1992) Riemannian geometry. Birkhäuser, BostonzbMATHCrossRefGoogle Scholar
- 15.Ferreira R, Xavier J, Costeira JP, Barroso V (2006) Newton method for Riemannian centroid computation in naturally reductive homogeneous spaces. In: ICASSPCrossRefGoogle Scholar
- 16.Fletcher PT (2013) Geodesic regression and the theory of least squares on riemannian manifolds. Int J Comput Vis 105(2):171–185zbMATHMathSciNetCrossRefGoogle Scholar
- 17.Fletcher PT, Lu C, Pizer SM, Joshi S (2004) Principal geodesic analysis for the study of nonlinear statistics of shape. Med. Imaging 23(8):995–1005CrossRefGoogle Scholar
- 18.Garrido L, Furl N, Draganski B, Weiskopf N, Stevens J, Chern-Yee Tan G, Driver J, Dolan RJ, Duchaine B (2009) Voxel-based morphometry reveals reduced grey matter volume in the temporal cortex of developmental prosopagnosics. Brain 132(12):3443–3455CrossRefGoogle Scholar
- 19.Goh A, Lenglet C, Thompson PM, Vidal R (2009) A nonparametric Riemannian framework for processing high angular resolution diffusion images (HARDI). In: CVPR, pp 2496–2503Google Scholar
- 20.Hardoon DR et al (2007) Unsupervised analysis of fMRI data using kernel canonical correlation. NeuroImage 37(4):1250–1259CrossRefGoogle Scholar
- 21.Hardoon DR, Szedmak S, Shawe-Taylor J (2004) CCA: an overview with application to learning methods. Neural Comput 16(12):2639–2664zbMATHCrossRefGoogle Scholar
- 22.Hinkle J, Fletcher PT, Joshi S (2014) Intrinsic polynomials for regression on Riemannian manifolds. J Math Imaging Vis 50:1–21MathSciNetCrossRefGoogle Scholar
- 23.Ho J, Xie Y, Vemuri B (2013) On a nonlinear generalization of sparse coding and dictionary learning. In: ICML, pp 1480–1488Google Scholar
- 24.Ho J, Cheng G, Salehian H, Vemuri B (2013) Recursive Karcher expectation estimators and geometric law of large numbers. In: Proceedings of the Sixteenth international conference on artificial intelligence and statistics, pp 325–332Google Scholar
- 25.Hotelling H (1936) Relations between two sets of variates. Biometrika 28(3/4):321–377zbMATHCrossRefGoogle Scholar
- 26.Hsieh WW (2000) Nonlinear canonical correlation analysis by neural networks. Neural Netw 13(10):1095–1105CrossRefGoogle Scholar
- 27.Hua X, Leow AD, Parikshak N, Lee S, Chiang MC, Toga AW, Jack CR Jr, Weiner MW, Thompson PM (2008) Tensor-based morphometry as a neuroimaging biomarker for Alzheimer’s disease: an MRI study of 676 AD, MCI, and normal subjects. NeuroImage 43(3):458–469CrossRefGoogle Scholar
- 28.Hua X, Gutman B et al (2011) Accurate measurement of brain changes in longitudinal MRI scans using tensor-based morphometry. NeuroImage 57(1):5–14CrossRefGoogle Scholar
- 29.Huang H, He H, Fan X, Zhang J (2010) Super-resolution of human face image using canonical correlation analysis. Pattern Recogn 43(7):2532–2543zbMATHCrossRefGoogle Scholar
- 30.Huckemann S, Hotz T, Munk A (2010) Intrinsic shape analysis: geodesic PCA for Riemannian manifolds modulo isometric Lie group actions. Stat Sin 20:1–100zbMATHMathSciNetGoogle Scholar
- 31.Jayasumana S, Hartley R, Salzmann M, Li H, Harandi M (2013) Kernel methods on the Riemannian manifold of symmetric positive definite matrices. In: CVPR, pp 73–80Google Scholar
- 32.Karcher H (1977) Riemannian center of mass and mollifier smoothing. Commun Pure Appl Math 30(5):509–541zbMATHMathSciNetCrossRefGoogle Scholar
- 33.Kim HJ, Adluru N, Collins MD, Chung MK, Bendlin BB, Johnson SC, Davidson RJ, Singh V (2014) Multivariate general linear models (MGLM) on Riemannian manifolds with applications to statistical analysis of diffusion weighted images. In: CVPRCrossRefGoogle Scholar
- 34.Kim T-K, Cipolla R (2009) CCA of video volume tensors for action categorization and detection. PAMI 31(8):1415–1428CrossRefGoogle Scholar
- 35.Klein A, Andersson J, Ardekani B, Ashburner J, Avants BB, Chiang M, Christensen G, Collins L, Hellier P, Song J, Jenkinson M, Lepage C, Rueckert D, Thompson P, Vercauteren T, Woods R, Mann J, Parsey R (2009) Evaluation of 14 nonlinear deformation algorithms applied to human brain MRI registration. NeuroImage 46:786–802CrossRefGoogle Scholar
- 36.Lai PL, Fyfe C (1999) A neural implementation of canonical correlation analysis. Neural Netw 12(10):1391–1397CrossRefGoogle Scholar
- 37.Lebanon G (2005) Riemannian geometry and statistical machine learning. PhD thesis, Carnegie Mellon UniversityGoogle Scholar
- 38.Li P, Wang Q, Zuo W, Zhang L (2013) Log-Euclidean kernels for sparse representation and dictionary learning. In: ICCV, pp 1601–1608Google Scholar
- 39.Moakher M (2005) A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J Matrix Anal Appl 26(3):735–747zbMATHMathSciNetCrossRefGoogle Scholar
- 40.Mostow GD (1973) Strong rigidity of locally symmetric spaces, vol 78. Princeton University Press, PrincetonzbMATHGoogle Scholar
- 41.Niethammer M, Huang Y, Vialard F (2011) Geodesic regression for image time-series. In: MICCAI, pp 655–662Google Scholar
- 42.Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
- 43.Rapcsák T (1991) Geodesic convexity in nonlinear optimization. J Opt Theory Appl 69(1):169–183zbMATHCrossRefGoogle Scholar
- 44.Said S, Courty N, Le Bihan N, Sangwine SJ et al (2007) Exact principal geodesic analysis for data on SO(3). In: Proceedings of the 15th European signal processing conference, pp 1700–1705Google Scholar
- 45.Shi X, Styner M, Lieberman J, Ibrahim JG, Lin W, Zhu H (2009) Intrinsic regression models for manifold-valued data. In: MICCAI, pp 192–199Google Scholar
- 46.Smith SM, Jenkinson M, Woolrich MW, Beckmann CF, Behrens TE, Johansen-Berg H, Bannister PR, De Luca M, Drobnjak I, Flitney DE, Niazy RK, Saunders J, Vickers J, Zhang Y, De Stefano N, Brady JM, Matthews PM (2004) Advances in functional and structural MR image analysis and implementation as FSL. NeuroImage 23:208–219CrossRefGoogle Scholar
- 47.Sommer S (2013) Horizontal dimensionality reduction and iterated frame bundle development. In: Geometric science of information. Springer, Heidelberg, pp 76–83CrossRefGoogle Scholar
- 48.Sommer S, Lauze F, Nielsen M (2014) Optimization over geodesics for exact principal geodesic analysis. Adv Comput Math 40(2):283–313zbMATHMathSciNetCrossRefGoogle Scholar
- 49.Steinke F, Hein M, Schölkopf B (2010) Nonparametric regression between general Riemannian manifolds. SIAM J Imaging Sci 3(3):527–563zbMATHMathSciNetCrossRefGoogle Scholar
- 50.Taniguchi H et al (1984) A note on the differential of the exponential map and jacobi fields in a symmetric space. Tokyo J Math 7(1):177–181zbMATHMathSciNetCrossRefGoogle Scholar
- 51.Witten DM, Tibshirani R, Hastie T (2009) A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics 10(3):515–534CrossRefGoogle Scholar
- 52.Xie Y, Vemuri BC, Ho J (2010) Statistical analysis of tensor fields. In: MICCAI, pp 682–689Google Scholar
- 53.Yger F, Berar M, Gasso G, Rakotomamonjy A (2012) Adaptive canonical correlation analysis based on matrix manifolds. In: ICMLGoogle Scholar
- 54.Yu S, Tan T, Huang K, Jia K, Wu X (2009) A study on gait-based gender classification. IEEE Trans Image Process 18(8):1905–1910MathSciNetCrossRefGoogle Scholar
- 55.Zhang H, Yushkevich PA, Alexander DC, Gee JC (2006) Deformable registration of diffusion tensor MR images with explicit orientation optimization. Med Image Anal 10:764–785CrossRefGoogle Scholar
- 56.Zhang Y, Brady M, Smith S (2001) Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximization algorithm. IEEE Trans Med Imging 20(1):45–57CrossRefGoogle Scholar
- 57.Zhu H, Chen Y, Ibrahim JG, Li Y, Hall C, Lin W (2009) Intrinsic regression models for positive-definite matrices with applications to diffusion tensor imaging. J Am Stat Assoc 104(487):1203–1212MathSciNetCrossRefGoogle Scholar