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Mapping Molecular Diffusion in the Plasma Membrane by Multiple-Target Tracing (MTT)
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Published on: May 27, 2012

Orientability and Diffusion Maps.

Amit Singer1, Hau-Tieng Wu

  • 1Department of Mathematics and PACM, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000 USA.

Applied and Computational Harmonic Analysis
|July 19, 2011
PubMed
Summary
This summary is machine-generated.

This study introduces an algorithm to determine manifold orientability in high-dimensional data. If orientable, it aids in Laplacian eigenfunction computation for diffusion maps; otherwise, it modifies diffusion mapping for the double covering.

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Area of Science:

  • Data Science
  • Computational Geometry
  • Topology

Background:

  • High-dimensional data analysis often assumes data lies on a low-dimensional manifold.
  • Understanding the geometric and topological structure of this manifold is crucial for effective analysis.

Purpose of the Study:

  • To introduce an algorithm for determining the orientability of intrinsic manifolds in high-dimensional data.
  • To provide methods for dimensionality reduction using diffusion maps, adapting for orientable and non-orientable manifolds.

Main Methods:

  • Algorithm development for manifold orientability detection.
  • Computation of Laplacian eigenfunctions for orientable manifolds.
  • Modified diffusion mapping for non-orientable manifolds via their orientable double covering.

Main Results:

  • Successfully determines manifold orientability from sampled data points.
  • Offers an alternative method for computing Laplacian eigenfunctions when the manifold is orientable.
  • Provides a modified diffusion mapping approach for non-orientable manifolds.

Conclusions:

  • The developed algorithm effectively characterizes manifold structure, crucial for dimensionality reduction.
  • The methods enhance the applicability of diffusion maps to a wider range of data structures, including non-orientable manifolds.