page_title: Nonlinear dimensionality reduction
hierarchical_section_title: Nonlinear dimensionality reduction
caption_attribution_description: Swissroll manifold recovery by the LLE and Hessian LLE algorithms
context_page_description: High-dimensional data, meaning data that requires more than two or three dimensions to represent, can be difficult to interpret. One approach to simplification is to assume that the data of interest lie on an embedded non-linear manifold within the higher-dimensional space. If the manifold is of low enough dimension, the data can be visualised in the low-dimensional space.
  Below is a summary of some of the important algorithms from the history of manifold learning and nonlinear dimensionality reduction. Many of these non-linear dimensionality reduction methods are related to the linear methods listed below. Non-linear methods can be broadly classified into two groups: those that provide a mapping, and those that just give a visualisation. In the context of machine learning, mapping methods may be viewed as a preliminary feature extraction step, after which pattern recognition algorithms are applied. Typically those that just give a visualisation are based on proximity data – that is, distance measurements.
context_section_description: High-dimensional data, meaning data that requires more than two or three dimensions to represent, can be  difficult to interpret. One approach to simplification is to assume that the data of interest lie on an embedded non-linear manifold within the higher-dimensional space. If the manifold is of low enough dimension, the data can be visualised in the low-dimensional space.
  Below is a summary of some of the important algorithms from the history of manifold learning and nonlinear dimensionality reduction (NLDR). Many of these non-linear dimensionality reduction methods are related to the linear methods listed below. Non-linear methods can be broadly classified into two groups: those that provide a mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa), and those that just give a visualisation.  In the context of machine learning, mapping methods may be viewed as a preliminary feature extraction step, after which pattern recognition algorithms are applied. Typically those that just give a visualisation are based on proximity data – that is, distance measurements.
caption: Top-left: a 3D dataset of 1000 points in a spiraling band (a.k.a. the Swiss roll) with a rectangular hole in the middle. Top-right: the original 2D manifold used to generate the 3D dataset. Bottom left and right: 2D recoveries of the manifold respectively using the LLE and Hessian LLE algorithms as implemented by the Modular Data Processing toolkit.
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