Abstract
This paper develops a supervised dimensionality reduction method, Lorentzian Discriminant Projection (LDP), for discriminant analysis and classification. Our method represents the structures of sample data by a manifold, which is furnished with a Lorentzian metric tensor. Different from classic discriminant analysis techniques, LDP uses distances from points to their within-class neighbors and global geometric centroid to model a new manifold to detect the intrinsic local and global geometric structures of data set. In this way, both the geometry of a group of classes and global data structures can be learnt from the Lorentzian metric tensor. Thus discriminant analysis in the original sample space reduces to metric learning on a Lorentzian manifold. The experimental results on benchmark databases demonstrate the effectiveness of our proposed method.
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References
Turk, M.A., Pentland, A.P.: Face Recognition Using Eigenfaces. In: Proc. CVPR (1991)
Belhumeur, P., Hespanha, J., Kriegman, D.: Eigenface vs. Fisherfaces: Recognition Using Class Specific Linear Projection. IEEE TPAMI 19, 711–720 (1997)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983)
Zhao, D., Lin, Z., Tang, X.: Classification via Semi-Riemannian Spaces. In: Proc. CVPR (2008)
Donoho, D.L., Grimes, C.: Hessian Eigenmaps: New Local Linear Embdedding Techniques for High-Dimensioanl Data. PNAS 102, 7426–7431 (2005)
Roweis, S.T., Saul, L.K.: Nonlinear Dimensionality Reduction by Locally Linear Embedding. Science 290, 2323–2326 (2000)
Tenenbaum, J.B., Silva, V.D., Langford, J.C.: A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science 290, 2319–2323 (2000)
Yan, S., Xu, D., Zhang, B., Zhang, H.J., Yang, Q., Lin, S.: Graph Embedding and Extensions: A General Framework for Dimensionality Reduction. IEEE TPAMI 27, 40–51 (2007)
He, X., Yan, S., Hu, Y., Niyogi, P., Zhang, H.J.: Face Recognition Using Laplacianfaces. IEEE TPAMI 27 (2005)
Zhang, Z., Zha, H.: Principal Manifolds and Nonlinear Dimensionality Reduction by Local Tangent Space Alignment. SIAM Journal of Scientific Computing 26, 313–338 (2004)
Lafon, S., Lee, A.B.: Diffusion Maps and Coarse-Graining: A Unified Framework for Dimensionality Reduction, Graph Partitioning, and Data Set Parameterization. IEEE TPAMI 28, 1393–1403 (2006)
Zhao, D.: Formulating LLE Using Alignment Technique. Pattern Recogition 39, 2233–2235 (2006)
Zhao, D., Lin, Z., Tang, X.: Laplacian PCA and Its Applications. In: Proc. ICCV (2007)
Belkin, M., Niyogi, P.: Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering. In: Proc. NIPS (2001)
He, X., Cai, D., Yan, S., Zhang, H.J.: Neighborhood Preserving Embedding. In: Proc. ICCV (2005)
He, X., Niyogi, P.: Locality Preserving Projections. In: Proc. NIPS (2003)
Li, H., Jiang, T., Zhang, K.: Efficient Robust Feature Extraction by Maximum Margin Criterion. In: Proc. NIPS (2003)
Wang, F., Zhang, C.: Feature Extraction by Maximizing the Average Neighborhood Margin. In: Proc. CVPR (2007)
Yang, J., Zhang, D., Yang, J.Y., Niu, B.: Globally Maximizing, Locally Minimizing: Unsupervised Discriminant Projection with Applications to Face and Palm Biometrics. IEEE TPAMI 29, 650–664 (2007)
Phillips, P.J., Flynn, P.J., Scruqqs, T., Bowyer, K.W., Jin, C., Hoffman, K., Marques, J., Jaesik, M., Worek, W.: Overview of The Face Recognition Grand Challenge. In: Proc. CVPR (2005)
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Liu, R., Su, Z., Lin, Z., Hou, X. (2010). Lorentzian Discriminant Projection and Its Applications. In: Zha, H., Taniguchi, Ri., Maybank, S. (eds) Computer Vision – ACCV 2009. ACCV 2009. Lecture Notes in Computer Science, vol 5996. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12297-2_30
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DOI: https://doi.org/10.1007/978-3-642-12297-2_30
Publisher Name: Springer, Berlin, Heidelberg
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