CSDL Home IEEE Transactions on Pattern Analysis & Machine Intelligence 2009 vol.31 Issue No.04 - April

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Issue No.04 - April (2009 vol.31)

pp: 749-754

Heng Lian , Nanyang Technological University, Singapore

ABSTRACT

We propose a novel model for nonlinear dimension reduction motivated by the probabilistic formulation of principal component analysis. Nonlinearity is achieved by specifying different transformation matrices at different locations of the latent space and smoothing the transformation using a Markov random field type prior. The computation is made feasible by the recent advances in sampling from von Mises-Fisher distributions. The computational properties of the algorithm are illustrated through simulations as well as an application to handwritten digits data.

INDEX TERMS

Statistical computing, Statistical

CITATION

Heng Lian, "Bayesian Nonlinear Principal Component Analysis Using Random Fields",

*IEEE Transactions on Pattern Analysis & Machine Intelligence*, vol.31, no. 4, pp. 749-754, April 2009, doi:10.1109/TPAMI.2008.212REFERENCES

- [1] T. Hastie, R. Tibshirani, and J.H. Friedman,
The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, 2001.- [2] J.S. Liu, J.L. Zhang, H. Palumbo, and C.E. Lawrence, “Bayesian Clustering with Variable and Transformation Selections (with Discussion),”
Bayesian Statistics, vol. 7, pp. 249-275, 2003.- [3] T. Hastie and W. Stuetzle, “Principal Curves,”
J. Am. Statistical Assoc., vol. 84, pp. 502-516, 1989.- [4] M.A. Kramer,
Probabilistic Principal Component Analysis Using Autoassociative Neural Networks, pp. 233-243, 1991.- [8] M.E. Tipping and C.M. Bishop, “Mixtures of Probabilistic Principal Component Analyzers,”
Neural Computation, vol. 11, no. 2, pp. 443-482, 1999.- [9] N. Lawrence, “Probabilistic Non-Linear Principal Component Analysis with Gaussian Process Latent Variable Models,”
J. Machine Learning Research, vol. 6, pp. 1783-1816, 2005.- [10] P. Hoff,
Simulation of the Matrix Bingham-Von Mises-Fisher Distribution, with Applications to Multivariate and Relational Data, http://www.citebase.orgabstract?id=oai:arXiv.org:0712.4166 , 2007.- [11] P. Hoff, “Model Averaging and Dimension Selection for the Singular Value Decomposition,”
J. Am. Statistical Assoc., vol. 102, pp. 674-685, 2007.- [12] S.Z. Li,
Markov Random Field Modeling in Computer Vision. Springer-Verlag, 1995.- [13] G. Winkler,
Image Analysis, Random Fields and Markov Chain Monte Carlo Methods: A Mathematical Introduction, second ed. Springer, 2003.- [14] C.M. Bishop and G.D. James, “Analysis of Multiphase Flows Using Dual-Energy Gamma Densitometry and Neural Networks,”
Nuclear Instruments and Methods in Physics Research, vol. 327, nos. 2-3, pp. 580-593, 1993.- [15] N. Lawrence, “Learning for Larger Data Sets with the Gaussian Process Latent Variable Model,”
Proc. 11th Int'l Conf. Artificial Intelligence and Statistics, 2007. |