Thursday, October 13, 2011, 1:00pm, French 4219

John Steenbergen

- 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 reduction.

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