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SCIENCE CHINA Information Sciences, Volume 61, Issue 9: 092104(2018) https://doi.org/10.1007/s11432-017-9233-4

Locality preserving projection on SPD matrix Lie group: algorithm and analysis

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  • ReceivedMar 15, 2017
  • AcceptedAug 16, 2017
  • PublishedApr 27, 2018

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.


Acknowledgment

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.


Supplement

Appendix A.


References

[1] Ma A J, Yuen P C, Zou W W W. Supervised Spatio-Temporal Neighborhood Topology Learning for Action Recognition. IEEE Trans Circuits Syst Video Technol, 2013, 23: 1447-1460 CrossRef Google Scholar

[2] 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. Google Scholar

[3] 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. Google Scholar

[4] 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. Google Scholar

[5] Wang L, Suter D. Learning and Matching of Dynamic Shape Manifolds for Human Action Recognition. IEEE Trans Image Process, 2007, 16: 1646-1661 CrossRef ADS Google Scholar

[6] Roweis S T. Nonlinear Dimensionality Reduction by Locally Linear Embedding. Science, 2000, 290: 2323-2326 CrossRef PubMed ADS Google Scholar

[7] Tenenbaum J B. A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science, 2000, 290: 2319-2323 CrossRef PubMed ADS Google Scholar

[8] He X F, Niyogi P. Locality preserving projections. In: Proceedings of the 16th Annual Conference on Neural Information Processing Systems, Chicago, 2003. Google Scholar

[9] 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. Google Scholar

[10] 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. Google Scholar

[11] Tuzel O, Porikli F, Meer P. Pedestrian detection via classification on Riemannian manifolds.. IEEE Trans Pattern Anal Mach Intell, 2008, 30: 1713-1727 CrossRef PubMed Google Scholar

[12] 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. Google Scholar

[13] 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. Google Scholar

[14] 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. Google Scholar

[15] 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. Google Scholar

[16] 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. Google Scholar

[17] 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. Google Scholar

[18] 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. Google Scholar

[19] Li Y Y. Locally preserving projection on symmetric positive definite matrix Lie group. In: Proceedings of the International Conference on Image Processing, Beijing, 2017. Google Scholar

[20] Arsigny V, Fillard P, Pennec X. Geometric Means in a Novel Vector Space Structure on Symmetric Positive?Definite Matrices. SIAM J Matrix Anal Appl, 2007, 29: 328-347 CrossRef Google Scholar

[21] 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. Google Scholar

[22] Willmore T. Riemannian Geometry. Oxford: Oxford University Press, 1997. Google Scholar

[23] Belkin M, Niyogi P. Towards a theoretical foundation for Laplacian-based manifold methods. J Comput Syst Sci, 2005, 74: 1289--1308. Google Scholar

[24] 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. Google Scholar

[25] Pennec X, Fillard P, Ayache N. A Riemannian Framework for Tensor Computing. Int J Comput Vision, 2006, 66: 41-66 CrossRef Google Scholar

[26] Muller M, Roder T, Clausen M, et al. Documentation: Mocap Database HDM05. Computer Graphics Technical Reports CG-2007-2. 2007. Google Scholar

[27] Smith L. A tutorial on principal components analysis. 2002,. arXiv Google Scholar

[28] Yale University. Face database. http://cvc.yale.edu/projects/yalefaces/yalefaces/html. Google Scholar

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