Nonlinear dimensionality reduction (NDR or NLDR)

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