Nonlinear dimensionality

Joshua B. Tenenbaum, Vin de Silva and John C. Langford, A global framework for nonlinear dimensionality reduction. Science, 290 (2000), 2319-2323. 

Sam T. Roweis and Lawrence K. Saul, Nonlinear dimensionality reduction by local linear embedding, Science, 290 (2000), 2323-2326. 

Lee JA, Verleysen M. Nonlinear Dimensionality Reduction. New York Springer 2007. 

Multidimensional scaling [98], nonlinear mapping [99], and other nonlinear dimensionality reduction approaches [100] intend to maintain pairwise distance or similarity relationships from the original high-dimensional space. This sometimes can better preserve close local relationships in a dataset, while detailed relationships between more dissimilar compounds are not maintained. 

Reutlinger M, Schneider G. Nonlinear dimensionality reduction and mapping of compound libraries for drug discovery. J Mol Graphics Model 2012 34 108—117. 

Livingstone et al. have employed a particular neural net architecture called a reversible nonlinear dimensionality reduction (ReNDeR) net for a low-dimensional display of multivariate data sets (160). The method makes use of the activity values of the hidden neurons in a trained three-layer feedforward network to produce the low-dimensional display. It was claimed that, in contrast to con-... 

Vandenberghe, L., Boyd, S.P. Semidefinite programming. SIAM Review 38(1), 49-95 (1996) Weinberger, K.Q., Sha, R, Saul, L.K. Learning a kernel matrix for nonlinear dimensionality reduction. In In Proceedings of the 21st International Conference on Machine Learning, pp. 839-846(2004)... 

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, 2319-2322 (2000)... 

Lee, J.A., Verleysen, M. Nonlinear dimensionality reduction of data manifolds with essential loops. Neurocomputing 67, 29-53 (2005)... 

As such, care should be taken when seeking to embed a dataset into higher dimensions. The performance of spectral dimensionality reduction methods, and in fact nonlinear dimensionality reduction methods in general, is called into question in such cases. 

Law, M., Jain, A. Incremental nonlinear dimensionality reduction by manifold learning. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(3), 377-391 (2006)... 

Zhang, K., Kwok, J.T. Clustered Nystrom Method for Laige Settle Manifold Learning and Dimension Reduction. IEEE Transactions on Neural Networks 21(10), 1576-1587 (2010) Silva, V.d., Tenenbaum, J.B. Global versus local methods in nonlinear dimensionality reduction. In Advances in Neural Information Rrocessing Systems 15 Proceedings of the 2003 Conference (NIPS), pp. 705-712. MIT Press (2003)... 

Zhang, X.L. Learning deep representation without parameter inference for nonlinear dimensionality reduction. ArXiv e-prints 1308.4922 (2013)... 

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