Nonlinear Dimension Reduction via Local Tangent Space Alignment

@inproceedings{Zhang2003NonlinearDR,
  title={Nonlinear Dimension Reduction via Local Tangent Space Alignment},
  author={Zhenyue Zhang and Hongyuan Zha},
  booktitle={IDEAL},
  year={2003}
}
In this paper we present a new algorithm for manifold learning and nonlinear dimension reduction. Based on a set of unorganized data points sampled with noise from the manifold, we represent the local geometry of the manifold using tangent spaces learned by fitting an affine subspace in a neighborhood of each data point. Those tangent spaces are aligned to give the internal global coordinates of the data points with respect to the underlying manifold by way of a partial eigendecomposition of… 
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References

SHOWING 1-10 OF 23 REFERENCES
Efficient Simplicial Reconstructions of Manifolds from Their Samples
  • D. Freedman
  • Mathematics
    IEEE Trans. Pattern Anal. Mach. Intell.
  • 2002
TLDR
An algorithm for manifold learning is presented and an important property of the algorithm is that its complexity depends on the dimension of the manifold, rather than that of the embedding space.
Nonlinear dimensionality reduction by locally linear embedding.
TLDR
Locally linear embedding (LLE) is introduced, an unsupervised learning algorithm that computes low-dimensional, neighborhood-preserving embeddings of high-dimensional inputs that learns the global structure of nonlinear manifolds.
Laplacian Eigenmaps for Dimensionality Reduction and Data Representation
TLDR
This work proposes a geometrically motivated algorithm for representing the high-dimensional data that provides a computationally efficient approach to nonlinear dimensionality reduction that has locality-preserving properties and a natural connection to clustering.
Global Coordination of Local Linear Models
TLDR
The regularizer takes the form of a Kullback-Leibler divergence and illustrates an unexpected application of variational methods: not to perform approximate inference in intractable probabilistic models, but to learn more useful internal representations in tractable ones.
Grouping and dimensionality reduction by locally linear embedding
TLDR
A variant of LLE that can simultaneously group the data and calculate local embedding of each group is studied, and an estimate for the upper bound on the intrinsic dimension of the data set is obtained automatically.
Stochastic Neighbor Embedding
TLDR
This probabilistic framework makes it easy to represent each object by a mixture of widely separated low-dimensional images, which allows ambiguous objects, like the document count vector for the word "bank", to have versions close to the images of both "river" and "finance" without forcing the image of outdoor concepts to be located close to those of corporate concepts.
Think globally, fit locally: unsupervised l earning of non-linear manifolds
A construction member having a tubular body with side walls surrounding a passage through the body. Each side wall has an opening and a plurality of notches for accommodating a connecting element.
Topology representing networks
Think globally...
TLDR
In an era of globalization, Google Earth and transcontinental air travel, all of us should learn a little about spherical geometry and its modern generalization, differential geometry, which underpins such imposing intellectual edifices as Einstein's general theory of relativity.
Applied Functional Data Analysis
TLDR
This book is a great book for a Ž rst graduate course in multivariate analysis, because it covers the standard topics in “classical” normal theory approach to multivariateAnalysis.
...
...