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Nonlinear dimensionality reduction (NDR or NLDR)

Nonlinear dimensionality reduction (NDR or NLDR)

A process of mapping higher-dimensional data into a lower-dimensional non-linear manifold within higher-dimensional space so that the data can be more easily visualized and interpreted.

Nonlinear dimensionality reduction (NDR or NLDR) is a process of mapping higher-dimensional data into a lower-dimensional non-linear manifold within higher-dimensional space so that the data can be more easily visualized and interpreted. In this context, a manifold is a mathematical space that -- when on a small enough scale -- resembles the Euclidean space of a specific dimension. Manifolds are useful in geometry and mathematical physics because they allow more complicated structures to be expressed and understood in terms of the relatively better-understood properties of simpler spaces.



NDR can be useful because variations in high-dimensional data often has much lower-dimensional explanations, and NDR can help researchers to visualize and understand the underlying structure of the data and the process that generated. 



There are two general methods of performing NDR:

  • Nonlinearize a linear dimensionality reduction method. (e.g. convert Kernel PCA into nonlinear PCA)
  • Use a manifold-based method.



Popular manifold-based methods for nonlinear dimensionality reduction include:



Timeline

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Further reading

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A Global Geometric Framework for Nonlinear Dimensionality Reduction

Joshua B. Tenenbaum, Vin de Silva, John C. Langford

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Nonlinear Dimensionality Reduction for Discriminative Analytics of Multiple Datasets

Jia Chen, Gang Wang, Georgios B. Giannakis

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On Nonlinear Dimensionality Reduction, Linear Smoothing and Autoencoding

Daniel Ting, Michael I. Jordan

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Documentaries, videos and podcasts

Title
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Lecture 21: Nonlinear Dimensionality Reduction

December 2, 2011

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