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Manifold

About: Manifold is a(n) research topic. Over the lifetime, 18786 publication(s) have been published within this topic receiving 362855 citation(s).

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Papers
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Open accessProceedings Article
Xiaofei He1, Partha Niyogi1Institutions (1)
09 Dec 2003-
Abstract: Many problems in information processing involve some form of dimensionality reduction. In this paper, we introduce Locality Preserving Projections (LPP). These are linear projective maps that arise by solving a variational problem that optimally preserves the neighborhood structure of the data set. LPP should be seen as an alternative to Principal Component Analysis (PCA) – a classical linear technique that projects the data along the directions of maximal variance. When the high dimensional data lies on a low dimensional manifold embedded in the ambient space, the Locality Preserving Projections are obtained by finding the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator on the manifold. As a result, LPP shares many of the data representation properties of nonlinear techniques such as Laplacian Eigenmaps or Locally Linear Embedding. Yet LPP is linear and more crucially is defined everywhere in ambient space rather than just on the training data points. This is borne out by illustrative examples on some high dimensional data sets.

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4,091 Citations


Open accessPosted Content
Abstract: UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology The result is a practical scalable algorithm that applies to real world data The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning

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Topics: Nonlinear dimensionality reduction (53%), Manifold (51%), Dimensionality reduction (51%) ...read more

3,034 Citations


Book ChapterDOI: 10.1007/978-1-4613-8101-3_1
Steve Smale1Institutions (1)
Abstract: This is a survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold M. An action is a homomorphism G→Diff(M) such that the induced map G×M→M is differentiable. Here Diff(M) is the group of all diffeomorphisms of M and a diffeo- morphism is a differentiable map with a differentiable inverse. Everything will be discussed here from the C ∞ or C r point of view. All manifolds maps, etc. will be differentiable (C r , 1 ≦ r ≦ ∞) unless stated otherwise.

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Topics: Differentiable function (67%), Periodic point (57%), Manifold (57%) ...read more

2,837 Citations


Open accessBook
01 Jun 1971-
Abstract: Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. It includes differentiable manifolds, tensors and differentiable forms. Lie groups and homogenous spaces, integration on manifolds, and in addition provides a proof of the de Rham theorem via sheaf cohomology theory, and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem. Those interested in any of the diverse areas of mathematics requiring the notion of a differentiable manifold will find this beginning graduate-level text extremely useful.

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Topics: Differential topology (69%), Symplectic geometry (64%), Manifold (64%) ...read more

1,891 Citations


Open accessBook
01 Jan 1975-
Abstract: This is a revised printing of one of the classic mathematics texts published in the last 25 years. This revised edition includes updated references and indexes and error corrections and will continue to serve as the standard text for students and professionals in the field.Differential manifolds are the underlying objects of study in much of advanced calculus and analysis. Topics such as line and surface integrals, divergence and curl of vector fields, and Stoke's and Green's theorems find their most natural setting in manifold theory. Riemannian plane geometry can be visualized as the geometry on the surface of a sphere in which "lines" are taken to be great circle arcs.

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Topics: Riemannian geometry (66%), Geometry and topology (63%), Differential geometry (62%) ...read more

1,873 Citations


Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202212
20211,033
20201,091
20191,037
20181,012
2017921

Top Attributes

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Topic's top 5 most impactful authors

Misha Verbitsky

48 papers, 794 citations

Liviu Ornea

32 papers, 682 citations

Uday Chand De

27 papers, 235 citations

Matti Lassas

21 papers, 296 citations

Gunther Uhlmann

20 papers, 847 citations

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