Using Graph Algebra to Optimize Neighborhood for Isometric Mapping

Most nonlinear dimensionality reduction approaches such as Isomap heavily depend on the neighborhood structure of manifold. They determine the neighborhood graph using Euclidean distance so that they often fail to nicely deal with sparsely sampled or noise contaminated data. This paper applies the graph algebra to optimize the neighborhood structure for Isomap. The improved Isomap outperforms the classic Isomap in visualization and time complexity, as it provides good neighborhood structure that can speed up the subsequent dimensionality reducing process. It also has stronger topological stability and less sensitive to parameters. This indicates that the more complicated or even time-consuming approaches can be applied to construct the better neighborhood structure whilst the whole time complexity will not raise. The conducted experiments on benchmark data sets have validated the proposed approach.

[1]  Li Yang.  Building k edge-disjoint spanning trees of minimum total length for isometric data embedding , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[2]  Matti Pietikäinen,et al.  Supervised Locally Linear Embedding , 2003, ICANN.

[3]  Matti Pietikäinen,et al.  Incremental locally linear embedding , 2005, Pattern Recognit..

[4]  Li Yang.  K-edge connected neighborhood graph for geodesic distance estimation and nonlinear data projection , 2004, ICPR 2004.

[5]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[6]  Daniel A. Spielman.  Graphs and networks , 2007 .

[7]  Amitabha Mukerjee,et al.  Non-linear Dimensionality Reduction by Locally Linear Isomaps , 2004, ICONIP.

[8]  Joachim M. Buhmann,et al.  Path-Based Clustering for Grouping of Smooth Curves and Texture Segmentation , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[9]  Houkuan Huang,et al.  Selection of the Optimal Parameter Value for the ISOMAP Algorithm , 2005, MICAI.

[10]  Mikhail Belkin,et al.  Laplacian Eigenmaps for Dimensionality Reduction and Data Representation , 2003, Neural Computation.

[11]  I. Anderson,et al.  Graphs and Networks , 1981, The Mathematical Gazette.

[12]  Matti Pietikäinen,et al.  Selection of the Optimal Parameter value for the Locally Linear Embedding Algorithm , 2002, FSKD.

[13]  Maja J. Mataric,et al.  A spatio-temporal extension to Isomap nonlinear dimension reduction , 2004, ICML.

[14]  Andrew Beng Jin Teoh,et al.  Fusion of Locally Linear Embedding and Principal Component Analysis for Face Recognition (FLLEPCA) , 2005, ICAPR.

[15]  S T Roweis,et al.  Nonlinear dimensionality reduction by locally linear embedding. , 2000, Science.

[16]  Anil K. Jain,et al.  Incremental nonlinear dimensionality reduction by manifold learning , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[17]  Changshui Zhang,et al.  Exploring the structure of supervised data by Discriminant Isometric Mapping , 2005, Pattern Recognit..

[18]  Joshua B. Tenenbaum,et al.  The Isomap Algorithm and Topological Stability , 2002, Science.

[19]  Joachim M. Buhmann,et al.  Bagging for Path-Based Clustering , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[20]  Dimitrios Gunopulos,et al.  Non-linear dimensionality reduction techniques for classification and visualization , 2002, KDD.

[21]  Li Yang,et al.  K-edge connected neighborhood graph for geodesic distance estimation and nonlinear data projection , 2004, Proceedings of the 17th International Conference on Pattern Recognition, 2004. ICPR 2004..

[22]  Zhi-Hua Zhou,et al.  Unified Locally Linear Embedding and Linear Discriminant Analysis Algorithm (ULLELDA) for Face Recognition , 2004, SINOBIOMETRICS.

[23]  Jian Xiao,et al.  Self-organized Locally Linear Embedding for Nonlinear Dimensionality Reduction , 2005, ICNC.

[24]  D. Donoho,et al.  Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[25]  Joshua B. Tenenbaum,et al.  Global Versus Local Methods in Nonlinear Dimensionality Reduction , 2002, NIPS.

[26]  Patrick Brézillon,et al.  Lecture Notes in Artificial Intelligence , 1999 .

[27]  Zhi-Hua Zhou,et al.  Supervised nonlinear dimensionality reduction for visualization and classification , 2005, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[28]  Xin Geng,et al.  Supervised nonlinear dimensionality reduction for visualization and classification , 2005, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).