Abstract
Locally linear embedding heavily depends on whether the neighborhood graph represents the underlying geometry structure of the data manifolds. Inspired from the cognitive relativity, this paper proposes a relative transformation that can be applied to build the relative space from the original space of data. In relative space, the noise and outliers will become further away from the normal points, while the near points will become relative closer. Accordingly we determine the neighborhood in the relative space for Hessian locally linear embedding, while the embedding is still performed in the original space. The conducted experiments on both synthetic and real data sets validate the approach.
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References
Roweis, S.T., Saul, L.K.: Nonlinear Dimensionality Reduction by Locally Linear Embedding. Science 290, 2323–2326 (2000)
Tenenbaum, J.B., de Silva, V., Langford, J.C.: A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science 290, 2319–2323 (2000)
Balasubramanian, M., Schwartz, E.L.: The ISOMAP Algorithm and Topological Stability. Science 295, 7 (2002)
Belkin, M., Niyogi, P.: Laplacian Eigenmaps for Dimensionality Reduction and Data Representation. Neural Computing 15, 1373–1396 (2003)
Donoho, D.L., Grimes, C.: Hessian eigenmaps: Locally linear embedding, techniques for high-dimensional data. Proc. Natl. Acad. Sci. 100, 5591–5596 (2003)
Silva, V.D., Tenenbaum, J.B.: Global versus local methods in nonlinear dimensionality reduction. Neural Information Processing Systems 15, 705–712 (2003)
Kouropteva, O., Okun, O., Pietikainen, M.: Incremental locally linear embedding. Pattern Recognition 38, 1764–1767 (2005)
Law, M.H.C., Jain, A.K.: Incremental nonlinear dimensionality reduction by manifold learning. IEEE Transactions on Pattern Analysis and Machine Intelligence 28, 377–391 (2006)
Samko, O., Marshall, A.D., Rosin, P.L.: Selection of the optimal parameter value for the Isomap algorithm. Pattern Recognition Letters 27(9), 968–979 (2006)
de Ridder, D., Kouropteva, O., Okun, O., et al.: Supervised locally linear embedding. In: Kaynak, O., Alpaydın, E., Oja, E., Xu, L. (eds.) ICANN 2003 and ICONIP 2003. LNCS, vol. 2714, pp. 333–341. Springer, Heidelberg (2003)
Chang, H., Yeung, D.-Y.: Robust locally linear embedding. Pattern Recognition 39, 1053–1065 (2006)
Yang, L.: Building k-Connected Neighborhood Graphs for Isometric Data Embedding. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(5), 827–831 (2006)
Yang, L.: Building Connected Neighborhood Graphs for Locally Linear Embedding. In: The 18th International Conference on Pattern Recognition, pp. 1680–1683 (2006)
de Ridder, D., Loog, M., Reinders, M.J.T.: Local Fisher embedding. In: Proceedings of the 17th International Conference on Pattern Recognition, pp. 295–298 (2004)
Saxena, A., Gupta, A., Mukerjee, A.: Non-linear dimensionality reduction by locally linear ISOMAPs. In: Pal, N.R., Kasabov, N., Mudi, R.K., Pal, S., Parui, S.K. (eds.) ICONIP 2004. LNCS, vol. 3316, pp. 1038–1043. Springer, Heidelberg (2004)
Sung, H.S., Lee, D.D.: The Manifold Ways of Perception. Science 290, 2268–2268 (2000)
Choi, H., Choi, S.: Robust kernel ISOMAP. Pattern Recognition 40, 853–862 (2007)
Li, D., Liu, C., Du, Y., Han, X.: Artificial Intelligence with Uncertainty. Journal of Software 15, 1583–1594 (2004)
Wen, G., Jiang, L., Wen, J., Shadbolt, N.R.: Performing Locally Linear Embedding with Adaptive Neighborhood Size on Manifold. In: Yang, Q., Webb, G. (eds.) PRICAI 2006. LNCS (LNAI), vol. 4099, pp. 985–989. Springer, Heidelberg (2006)
Wen, G., Jiang, L., Wen, J., Shadbolt, N.R.: Clustering-based Nonlinear Dimensionality Reduction on Manifold. In: Yang, Q., Webb, G. (eds.) PRICAI 2006. LNCS (LNAI), vol. 4099, pp. 444–453. Springer, Heidelberg (2006)
Wen, G., Jiang, L., Shadbolt, N.R.: Using Graph Algebra to Optimize Neighborhood for Isometric Mapping. In: 20th International Joint Conference on Artificial Intelligence(IJCAI-07), India, pp. 2398–2403 (2007)
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Wen, G., Jiang, L., Wen, J. (2007). Improved Locally Linear Embedding by Cognitive Geometry. In: Li, K., Li, X., Irwin, G.W., He, G. (eds) Life System Modeling and Simulation. LSMS 2007. Lecture Notes in Computer Science(), vol 4689. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74771-0_36
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DOI: https://doi.org/10.1007/978-3-540-74771-0_36
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74770-3
Online ISBN: 978-3-540-74771-0
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