Data Seminar Seminar
Thursday, October 13, 2011, 1:00pm, French 4219
Dimension Reduction, Laplacians, and Cheeger Numbers
- The graph Laplacian is used by the Laplacian Eigenmaps algorithm to
perform dimension reduction on data. Many other dimension reduction
methods (LLE, Hessian LLE, and diffusion maps) bear some relation to the
Laplacian as well. What is it that makes the graph Laplacian so useful?
This question has in the past been answered by relating the graph
Laplacian to the Cheeger number and to the Laplace-Beltrami operator on
manifolds (which itself relates to the continuous Cheeger number on
manifolds). Given that graphs are special cases of simplicial complexes
and that the graph Laplacian is just one of the many combinatorial
Laplacians (one for each dimension), we are led to the following question.
Can this framework of Laplacians and Cheeger numbers, and Laplacian-based
dimension reduction methods themselves, be 'scaled' to higher dimensions?
We introduce some recent research in this direction and roughly discuss
what it might mean to use higher-order Laplacians to perform dimension
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