Editing Nonlinear dimensionality reduction (section)

{{Short description|Projection of data onto lower-dimensional manifolds}}
[[File:Lle hlle swissroll.png|thumb|right|300px|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 [[Nonlinear dimensionality reduction#Locally-linear embedding|LLE]] and [[Nonlinear dimensionality reduction#Hessian Locally-Linear Embedding (Hessian LLE)|Hessian LLE]] algorithms as implemented by the Modular Data Processing toolkit.]]

'''Nonlinear dimensionality reduction''', also known as '''manifold learning''', is any of various related techniques that aim to project high-dimensional data, potentially existing across non-linear manifolds which cannot be adequately captured by linear decomposition methods, onto lower-dimensional [[latent manifold]]s, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa) itself.<ref>{{cite journal|last=Lawrence|first=Neil D|title=A unifying probabilistic perspective for spectral dimensionality reduction: insights and new models|year=2012|pages=1609–38|volume=13|issue=May|journal=[[Journal of Machine Learning Research]]|url=http://www.jmlr.org/papers/v13/lawrence12a.html|bibcode=2010arXiv1010.4830L|arxiv=1010.4830}}</ref><ref>{{cite book |first1=John A. |last1=Lee |first2=Michel |last2=Verleysen |title=Nonlinear Dimensionality Reduction |publisher=Springer |year=2007 |isbn=978-0-387-39350-6 }}</ref> The techniques described below can be understood as generalizations of linear decomposition methods used for [[dimensionality reduction]], such as [[singular value decomposition]] and [[principal component analysis]].