A flag representation for finite collections of subspaces of mixed dimensions. (English) Zbl 1326.14118
Summary: Given a finite set of subspaces of \(\mathbb{R}^n\), perhaps of differing dimensions, we describe a flag of vector spaces (i.e. a nested sequence of vector spaces) that best represents the collection based on a natural optimization criterion and we present an algorithm for its computation. The utility of this flag representation lies in its ability to represent a collection of subspaces of differing dimensions. When the set of subspaces all have the same dimension \(d\), the flag mean is related to several commonly used subspace representations. For instance, the \(d\)-dimensional subspace in the flag corresponds to the extrinsic manifold mean. When the set of subspaces is both well clustered and equidimensional of dimension \(d\), then the \(d\)-dimensional component of the flag provides an approximation to the Karcher mean. An intermediate matrix used to construct the flag can also be used to recover the canonical components at the heart of Multiset Canonical Correlation Analysis. Two examples utilizing the Carnegie Mellon University Pose, Illumination, and Expression Database (CMU-PIE) serve as visual illustrations of the algorithm.
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 53B20 | Local Riemannian geometry |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 62B10 | Statistical aspects of information-theoretic topics |
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 94B15 | Cyclic codes |
Keywords:
subspace average; Grassmann manifold; flag manifold; SVD; flag mean; multiset canonical correlation analysis; extrinsic manifold meanSoftware:
CMU PIEReferences:
| [1] | Begelfor, E.; Werman, M., Affine invariance revisited, (Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 2. Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 2, CVPR ’06 (2006), IEEE Computer Society: IEEE Computer Society Washington, DC, USA), 2087-2094 |
| [2] | Belhumeur, P. N.; Kriegman, D. J., What is the set of images of an object under all possible illumination conditions?, Int. J. Comput. Vis., 28, 3, 245-260 (1998) |
| [3] | Björck, A.; Golub, G. H., Numerical methods for computing angles between linear subspaces, Math. Comp., 27, 579-594 (1973) · Zbl 0282.65031 |
| [4] | Chang, J. M.; Kirby, M.; Kley, H.; Peterson, C.; Draper, B.; Beveridge, J. R., Recognition of digital images of the human face at ultra low resolution via illumination spaces, (Proceedings of the 8th Asian Conference on Computer Vision-Volume, Part II (2007), Springer-Verlag), 733-743 |
| [5] | Edelman, A.; Arias, T.; Smith, S. T., The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20, 303-353 (1998) · Zbl 0928.65050 |
| [6] | Hamm, J., Subspace-based learning with Grassmann kernels (2008), University of Pennsylvania, PhD thesis |
| [7] | Karcher, H., Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math., 30, 5, 509-541 (1977) · Zbl 0354.57005 |
| [8] | Kettenring, J. R., Canonical analysis of several sets of variables, Biometrika, 58, 3, 433-451 (1971) · Zbl 0225.62072 |
| [9] | Kirby, Michael; Sirovich, Lawrence, Application of the Karhunen-Loeve procedure for the characterization of human faces, IEEE Trans. Pattern Anal. Mach. Intell., 12, 1, 103-108 (1990) |
| [10] | Lui, Y. M.; Beveridge, J. R.; Kirby, M., Action classification on product manifolds, (2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (June 2010)), 833-839 |
| [11] | Moakher, Maher, Means and averaging in the group of rotations, SIAM J. Matrix Anal. Appl., 24, 1, 1-16 (2002) · Zbl 1028.47014 |
| [12] | Mondal, B.; Dutta, S.; Heath, R. W., Quantization on the Grassmann manifold, IEEE Trans. Signal Process., 55, 8, 4208-4216 (2007) · Zbl 1390.94309 |
| [13] | Monk, D., The geometry of flag manifolds, Proc. Lond. Math. Soc., 3, 2, 253-286 (1959) · Zbl 0096.36201 |
| [14] | Patrangenaru, V.; Mardia, K. V., Affine shape analysis and image analysis, (Proc. 22nd Leeds Ann. Statistics Research Workshop (July 2003)) |
| [15] | Sarlette, Alain; Sepulchre, Rodolphe, Consensus optimization on manifolds, SIAM J. Control Optim., 48, 1, 56-76 (2009) · Zbl 1182.93010 |
| [16] | Sim, T.; Baker, S.; Bsat, M., The CMU pose, illumination, and expression (PIE) database, (Proceedings of the 5th International Conference on Automatic Face and Gesture Recognition (2002)) |
| [17] | Srivastava, A.; Klassen, E., Monte Carlo extrinsic estimators of manifold-valued parameters, IEEE Trans. Signal Process., 50, 2, 299-308 (2002) |
| [18] | Srivastava, A.; Klassen, E., Bayesian and geometric subspace tracking, Adv. in Appl. Probab., 36, 1, 43-56 (2004) · Zbl 1047.93044 |
| [19] | Turaga, P.; Veeraraghavan, A.; Srivastava, A.; Chellappa, R., Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition, IEEE Trans. Pattern Anal. Mach. Intell., 33, 11, 2273-2286 (2011) |
| [20] | Tuzel, O.; Porikli, F.; Meer, P., Human detection via classification on Riemannian manifolds, (Proc. IEEE Conf. Computer Vision and Pattern Recognition (CVPR) (2007)), 1-8 |
| [21] | Via, J.; Santamaria, I.; Perez, J., A learning algorithm for adaptive canonical correlation analysis of several data sets, Neural Networks, 20, 1, 139-152 (2007) · Zbl 1158.68459 |
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