Abstract
Symmetric positive definite (SPD) matrices used as feature descriptors in image recognition are usually high dimensional. Traditional manifold learning is only applicable for reducing the dimension of high-dimensional vector-form data. For high-dimensional SPD matrices, directly using manifold learning algorithms to reduce the dimension of matrix-form data is impossible. The SPD matrix must first be transformed into a long vector, and then the dimension of this vector must be reduced. However, this approach breaks the spatial structure of the SPD matrix space. To overcome this limitation, we propose a new dimension reduction algorithm on SPD matrix space to transform high-dimensional SPD matrices into low-dimensional SPD matrices. Our work is based on the fact that the set of all SPD matrices with the same size has a Lie group structure, and we aim to transform the manifold learning to the SPD matrix Lie group. We use the basic idea of the manifold learning algorithm called locality preserving projection (LPP) to construct the corresponding Laplacian matrix on the SPD matrix Lie group. Thus, we call our approach Lie-LPP to emphasize its Lie group character. We present a detailed algorithm analysis and show through experiments that Lie-LPP achieves effective results on human action recognition and human face recognition.
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References
Ma A J, Yuen P C, Zou W W W, et al. Supervised spatio-temporal neighborhood topology learning for action recognition. IEEE Trans Circ Syst Video Technol, 2013, 23: 1447–1460
Hussein M E, Torki M, Gowayyed M A, et al. Human action recognition using a temporal hierarchy of covariance descriptors on 3D joint locations. In: Proceedings of the 23rd International Joint Conference on Artificial Intelligence, Beijing, 2013. 2466–2472
Vemulapalli R, Arrate F, Chellappa R. Human action recognition by representing 3D skeletons as points in a Lie group. In: Proceedings of the 2014 IEEE Conference on Computer Vision and Pattern Recognition, Columbus, 2014
Tuzel O, Porikli F, Meer P. Region covariance: a fast descriptor for detection and classification. In: Proceedings of the 7th European Conference on Computer Vision: Part II, Graz, 2006. 589–600
Wang L, Suter D. Learning and matching of dynamic shape manifolds for human action recognition. IEEE Trans Image Process, 2007, 16: 1646–1661
Roweis S, Saul L. Nonlinear dimensionality reduction by locally linear embedding. Science, 2000, 290: 2323–2326
Tenenbaum J B, Silva V, Langford J. A global geometric framework for nonlinear dimensionality reduction. Science, 2000, 290: 2319–2323
He X F, Niyogi P. Locality preserving projections. In: Proceedings of the 16th Annual Conference on Neural Information Processing Systems, Chicago, 2003
He X F, Cai D, Niyogi P. Tensor subspace analysis. In: Proceedings of the 18th Annual Conference on Neural Information Processing Systems, British Columbia, 2005. 499–506
Porikli F, Tuzel O. Fast construction of covariance matrices for arbitrary size image windows. In: Proceedings of the International Conference on Image Processing, Atlanta, 2006. 1581–1584
Tuzel O, Porikli F, Meer P. Pedestrian detection via classification on Riemannian manifolds. IEEE Trans PAMI, 2008, 30: 1713–1727
Porikli F, Tuzel O, Meer P. Covariance tracking using model update based on Lie algebra. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Washington, 2006
Sanin A, Sanderson C, Harandi M T, et al. Spatio-temporal covariance descriptors for action and gesture recognition. In: Proceedings of the 2013 IEEE Workshop on Applications of Computer Vision, Washington, 2013. 103–110
Tabia H, Laga H, Picard D, et al. Covariance descriptors for 3D shape matching and retrieval. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Columbus, 2014
Huang Z W, Wang R P, Shan S G, et al. Log-Euclidean metric learning on symmetric positive definite manifold with application to image set classification. In: Proceedings of the 32nd International Conference on Machine Learning, Lille, 2015. 720–729
Harandi M, Sanderson C, Wiliem A, et al. Kernel analysis over Riemannian manifolds for visual recognition of actions, pedestrians and textures. In: Proceedings of the IEEE Workshop on the Applications of Computer Vision, Breckenridge, 2012
Harandi M, Salzmann M, Hartley R. From manifold to manifold: geometry-aware dimensionality reduction for SPD matrices. In: Proceedings of the 15th European Conference on Computer Vision, Zurich, 2014. 17–32
Kwatra V, Han M. Fast covariance computation and dimensionality reduction for sub-window features in image. In: Proceedings of the 11th European Conference on Computer Vision: Part II, Heraklion, 2010. 156–169
Li Y Y. Locally preserving projection on symmetric positive definite matrix Lie group. In: Proceedings of the International Conference on Image Processing, Beijing, 2017
Arsigny V, Fillard P, Pennec X, et al. Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J Matrix Anal Appl, 2007, 29: 328–347
Belkin M, Niyogi P. Laplacian eigenmaps and spectral techniques for embedding and clustering. In: Proceedings of the International Conference Advances in Neural Information Processing Systems, Vancouver, 2001. 585–591
Willmore T. Riemannian Geometry. Oxford: Oxford University Press, 1997
Belkin M, Niyogi P. Towards a theoretical foundation for Laplacian-based manifold methods. J Comput Syst Sci, 2005, 74: 1289–1308
Li X, Hu W M, Zhang Z F, et al. Visual tracking via incremental Log-Euclidean Riemannian subspace learning. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Anchorage, 2008
Pennec X, Fillard P, Ayache N. A Riemannian framework for tensor computing. Int J Comput Vision, 2006, 66: 41–66
Muller M, Roder T, Clausen M, et al. Documentation: Mocap Database HDM05. Computer Graphics Technical Reports CG-2007-2. 2007
Smith L. A tutorial on principal components analysis. 2002. arXiv:1404.1100
Yale University. Face database. http://cvc.yale.edu/projects/yalefaces/yalefaces/html
Acknowledgements
This work was supported by National Key Research and Development Program of China (Grant No. 2016YFB1000902), National Natural Science Foundation of China (Grant Nos. 61232015, 61472412, 61621003), Beijing Science and Technology Project on Machine Learning-based Stomatology, and Tsinghua-Tencent-AMSS-Joint Project on WWW Knowledge Structure and its Application.
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Li, Y., Lu, R. Locality preserving projection on SPD matrix Lie group: algorithm and analysis. Sci. China Inf. Sci. 61, 092104 (2018). https://doi.org/10.1007/s11432-017-9233-4
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DOI: https://doi.org/10.1007/s11432-017-9233-4