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:

- Locally Linear Embedding (LLE)
- Isomap
- Maximum Variance Unfolding
- Laplacian Eigenmaps

### Timeline

### Further Resources

A Global Geometric Framework for Nonlinear Dimensionality Reduction

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

Journal

Lecture 21: Nonlinear Dimensionality Reduction

December 2, 2011

Nonlinear Dimensionality Reduction for Discriminative Analytics of Multiple Datasets

Jia Chen, Gang Wang, Georgios B. Giannakis

On Nonlinear Dimensionality Reduction, Linear Smoothing and Autoencoding

Daniel Ting, Michael I. Jordan