Geometry-aware principal component analysis for symmetric positive definite matrices

ACML 2015 - 7th Asian Conference on Machine Learning

Volumn , Issue , 2015, Pages 1-16

  Horev, Inbal ^a   Yger, Florian ^b   Sugiyama, Masashi ^b  

^a TOKYO INSTITUTE OF TECHNOLOGY   (Japan)

^b UNIVERSITY OF TOKYO   (Japan)

Author keywords

Dimensionality reduction; Grassmann manifold; PCA; Riemannian geometry; SPD manifold

Indexed keywords

ARTIFICIAL INTELLIGENCE; COVARIANCE MATRIX; DATA HANDLING; DATA REDUCTION; EDUCATION; GEOMETRY; LEARNING SYSTEMS; MATRIX ALGEBRA;

DIMENSIONALITY REDUCTION; DIMENSIONALITY REDUCTION TECHNIQUES; GRASSMANN MANIFOLD; MACHINE LEARNING APPLICATIONS; RIEMANNIAN GEOMETRY; SPD MANIFOLD; SYMMETRIC POSITIVE DEFINITE; SYMMETRIC POSITIVE DEFINITE MATRICES;

PRINCIPAL COMPONENT ANALYSIS;

EID: 85026322805     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (37)

1. Pierre-Antoine Absil, Robert Mahony, and Rodolphe Sepulchre. Optimization Algorithms on Matrix Manifolds. Princeton University Press, 2009.
2. Awad H. Al-Mohy and Nicholas J. Higham. Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation. SIAM Journal on Matrix Analysis and Applications., 30(4):1639-1657, 2009.
3. Vincent Arsigny Pierre Fillard, Xavier Pennec, and Nicholas Ayache. Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magnetic Resonance in Medicine., 56(2):411-421, 2006.
4. Vincent Arsigny, Pierre Fillard, Xavier Pennec, and Nicholas Ayache. Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM Journal on Matrix Analysis and Applications., 29(1):328-347, 2007.
5. Alexandre Barachant and Marco Congedo. A Plug&Play P300 BCI using information geometry. arXiv preprint arXiv:1409.0107, 2014.
6. Alexandre Barachant, Stéphane Bonnet, Marco Congedo, and Christian Jutten. Riemannian geometry applied to BCI classification. In Latent Variable Analysis and Signal Separation, pages 629-636, 2010.
7. Alexandre Barachant, Stéphane Bonnet, Marco Congedo, and Christian Jutten. Multiclass brain-computer interface classification by Riemannian geometry. IEEE Transactions on Biomedical Engineering, 59(4):920-928, 2012.
8. Alexandre Barachant, Stéphane Bonnet, Marco Congedo, and Christian Jutten. Classification of covariance matrices using a Riemannian-based kernel for BCI applications. Neurocomputing, 112: 172-178, 2013.
9. Rajendra Bhatia. Matrix Analysis. Springer, 1997.
10. Rajendra Bhatia. Positive Definite Matrices. Princeton University Press, 2009.
11. Benjamin Blankertz, Ryota Tomioka, Steven Lemm, Motoaki Kawanabe, and Klaus-Robert Müller. Optimizing spatial filters for robust EEG single-trial analysis. IEEE Signal Processing Magazine, 25(1):41-56, 2008.
12. Silvere Bonnabel. Stochastic gradient descent on Riemannian manifolds. IEEE Transactions on Automatic Control, 58(9):2217-2229, Sept 2013.
13. Nicolas Boumal. Discrete curve fitting on manifolds. Master's thesis, Université Catholique de Louvain, jun 2010.
14. Nicolas Boumal and Pierre-Antoine Absil. Discrete regression methods on the cone of positive-definite matrices. In IEEE International Conference on Acoustics, Speech and Signal Processing, pages 4232-4235, 2011.
15. Nicolas Boumal, Bamdev Mishra, Pierre-Antoine Absil, and Rodolphe Sepulchre. Manopt, a Matlab toolbox for optimization on manifolds. Journal of Machine Learning Research, 15:1455-1459, 2014. URL http://www.manopt.org.
16. Anoop Cherian and Suvrit Sra. Riemannian sparse coding for positive definite matrices. In European Conference on Computer Vision, pages 299-314, 2014.
17. Andrzej Cichocki, Sergio Cruces, and Shun-Ichi Amari. Log-Determinant divergences revisited: Alpha-beta and gamma log-det divergences. arXiv preprint arXiv:1412.7146, 2014.
18. Arthur P Dempster. Covariance selection. Biometrics, pages 157-175, 1972.
19. Alan Edelman, Tomás A. Arias, and Steven T. Smith. The geometry of algorithms with orthogonality constraints. SIAM Journal on Matrix Analysis and Applications, 20(2):303-353, 1998.
20. P. Thomas Fletcher, Conglin Lu, Stephen M. Pizer, and Sarang Joshi. Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions on Medical Imaging, 23(8): 995-1005, 2004.
21. Maurice Fréchet. Les éléments aléatoires de nature quelconque dans un espace distancié. In Annales de l'institut Henri Poincaré, Volume 10, pages 215-310. Presses Universitaires de France, 1948.
22. Mehrtash Harandi, Mathieu Salzmann, and Richard Hartley. From manifold to manifold: geometry-aware dimensionality reduction for SPD matrices. In European Conference on Computer Vision, pages 17-32, 2014.
23. Jeffrey Ho, Yuchen Xie, and Baba Vemuri. On a nonlinear generalization of sparse coding and dictionary learning. In International Conference on Machine Learning, pages 1480-1488, 2013.
24. Stephan Huckemann, Thomas Hotz, and Axel Munk. Intrinsic shape analysis: Geodesic PCA for Riemannian manifolds modulo isometric Lie group actions. Statistica Sinica, 20:1-100, 2010.
25. Ian Jolliffe. Principal Component Analysis. Springer, 2002.
26. Fabien Lotte and Cuntai Guan. Regularizing common spatial patterns to improve BCI designs: Unified theory and new algorithms. IEEE Transactions on Biomedical Engineering, 58(2):355-362, 2011.
27. Haiping Lu, Konstantinos N. Plataniotis, and Anastasios N. Venetsanopoulos. Multilinear principal component analysis of tensor objects for recognition. In International Conference on Pattern Recognition, Volume 2, pages 776-779, 2006.
28. Xavier Pennec, Pierre Fillard, and Nicholas Ayache. A Riemannian framework for tensor computing. International Journal of Computer Vision, 66(1):41-66, 2006.
29. Alois Schlögl, Felix Lee, Horst Bischof, and Gert Pfurtscheller. Characterization of four-class motor imagery EEG data for the BCI-competition 2005. Journal of Neural Engineering, 2(4):L14, 2005.
30. Stefan Sommer, François Lauze, Søren Hauberg, and Mads Nielsen. Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations. In European Conference on Computer Vision, pages 43-56, 2010.
31. Suvrit Sra. Positive definite matrices and the s-divergence. arXiv preprint arXiv:1110.1773, 2011.
32. Suvrit Sra. A new metric on the manifold of kernel matrices with application to matrix geometric means. In Neural Information Processing Systems, pages 144-152, 2012.
33. Oncel Tuzel, Fatih Porikli, and Peter Meer. Region covariance: A fast descriptor for detection and classification. In European Conference on Computer Vision, pages 589-600, 2006.
34. Oncel Tuzel, Fatih Porikli, and Peter Meer. Pedestrian detection via classification on Riemannian manifolds. IEEE Transactions on Pattern Analysis and Machine Intelligence, 30(10):1713-1727, 2008.
35. Jian Yang, David Zhang, Alejandro F. Frangi, and Jing-yu Yang. Two-dimensional PCA: A new approach to appearance-based face representation and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(1):131-137, 2004.
36. Florian Yger. A review of kernels on covariance matrices for BCI applications. In IEEE International Workshop on Machine Learning for Signal Processing, pages 1-6, 2013.
37. Florian Yger and Masashi Sugiyama. Supervised logEuclidean metric learning for symmetric positive definite matrices. preprint arXiv:1502.03505, 2015.