Abstract An automatic neighborhood selection algorithm for manifold learning is proposed. It is suitable for all the manifold learning algorithms which need select the neighbors to get locally linear information. Through the algorithm, the proper neighborhood size of a dataset can be determined even under different data density and curvature. By adopting this method, the locally multidimensional scaling can reduce the dimensionality of data based on the suitable neighborhood, and the low-dimensional representations of the data can be get through global alignment. The experiment shows the algorithm can recover the sophisticated geometry structure of the data sets.
[1] Tenenbaum J B, de Silva V, Langford J C. A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science, 2000, 290(5500): 2319-2323 [2] de Silva V, Tenenbaum J B. Global versus Local Methods in Nonlinear Dimensionality Reduction // Becker S, Thrun S, Obermayer K, eds. Advances in Neural Information Processing Systems. Cambridge, USA: MIT Press, 2002, 15: 705-712 [3] Roweis S T, Saul L K. Nonlinear Dimensionality Reduction by Locally Linear Embedding. Science, 2000, 290(5500): 2323-2326 [4] Saul L K, Roweis S T. Think Globally, Fit Locally: Unsupervised Learning of Low Dimensional Manifolds. Journal of Machine Learning Research, 2003, 4(6): 119-155 [5] Zhang Zhenyue, Zha Hongyuan. Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment. SIAM Journal of Scientific Computing, 2005, 26(1): 313-338 [6] Zhao Deli. Formulating LLE Using Alignment Technique. Pattern Recognition, 2006, 39(11): 2233-2235 [7] Wu Yiming, Chan K L. An Extended ISOMAP Algorithm for Learning Multi-Class Manifold // Proc of the 3rd International Conference on Machine Learning and Cybernetics. Shanghai, China, 2004, Ⅵ: 3429-3433 [8] Yan Jian, Li Fuxin, Wang Jue. A Better Scaled Local Tangent Space Alignment Algorithm. Journal of Software,2005,16(9): 1584-1590 (in Chinese) (杨 剑,李伏欣,王 珏.一种改进的局部切空间排列算法.软件学报, 2005, 16(9): 1584-1590) [9] Balasubramanian M, Schwartz E L, Tenenbaum J B, et al. The ISOMAP Algorithm and Topological Stability. Science, 2002, 295(5552): 7 [10] Yang Li. Locally Multidimensional Scaling for Nonlinear Dimensionality Reduction // Proc of the 18th International Conference on Pattern Recognition. Hongkong, China, 2006, Ⅳ: 202-205 [11] Luo Siwei, Zhao Lianwei. Manifold Learning Algorithm Based on Spectral Graph Theory. Journal of Computer Research and Development, 2006, 43(7): 1173-1179 (in Chinese) (罗四维,赵连伟.基于谱图理论的流形学习.计算机研究与发展, 2006, 43(7): 1173-1179) [12] Wang Jing. Research on Manifold Learning: Theories and Approaches. Ph.D Dissertation. Hangzhou, China: Zhejiang University. College of Computer Science and Technology, 2003 (in Chinese) (王 靖.流形学习的理论与方法研究.博士学位论文.杭州:浙江大学.计算机科学与技术学院, 2003) [13] Camastra F. Data Dimensionality Estimation Methods: A Survey. Pattern Recognition, 2003, 36(12): 2945-2954
2008, Vol. 21 
