Abstract A local linear space partition (LLSP) based manifold generation algorithm is proposed to extend the mapping function obtained by manifold learning of new data points. Using the dimension-fixed projection distance (DFPD), the dimension-fixed projection vector quantization (DFPVQ) algorithm is presented to cover the whole manifold with several local linear spaces (LLS). Then, the simplified linear mapping functions are constructed in LLS. As for new data point, the corresponding LLS is found and the mapping value in low dimension is estimated by the simplified linear mapping function. The superiority of the proposed algorithm is confirmed by experimental results both on synthesized data and handwritten digits image dataset.
[1] Jolliffe I T. Principal Component Analysis. New York, USA: Springer-Verlag, 1989 [2] Cox T F, Cox M A. Multidimensional Scaling. London, UK: Chapman and Hall, 1994 [3] Roweis S T, Saul L K. Nonlinear Dimensionality Reduction by Locally Linear Embedding. Science, 2000, 290(5500): 2323-2326 [4] Tenenbaum J B, vin de Silva, Langford J C. A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science, 2000, 290(5500): 2319-2323 [5] Belkin M, Niyogi P. Laplacian Eigenmaps for Dimensionality Reduction and Data Representation. Neural Computation, 2003, 15(6): 1373-1396 [6] Zhang Zhenyue, Zha Hongyuan. Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment. SIAM Journal of Scientific Computing, 2004, 26(1): 313-338 [7] Bengio Y, Paiement J F, Vincent P . Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering. Cambridge, USA: MIT Press, 2003 [8] Shi Lukui, He Pilian, Liu Bin, et al. A Robust Generalization of Isomap for New Data // Proc of the 4th International Conference on Machine Learning and Cybernetics. Guangzhou, China, 2005, Ⅲ: 1702-1712 [9] Olga K, Oleg O, Matti P. Incremental Locally Linear Embedding. Pattern Recognition, 2005, 38(10): 1764-1767 [10] Kambhatla N, Leen T K. Dimension Reduction by Local Principal Component Analysis. Neural Computation, 1997, 9(7): 1493-1516 [11] LeCun Y, Cortes C. MNIST Database[DB/OL]. [2007-12-15]. http://yann.lecun.com/exdb/mnist/
2009, Vol. 22 
