Data Seminar
Monday, November 12, 2012, 11:30am, Phys
John Steenbergen
A Cheeger-Type Inequality on Simplicial Complexes
Abstract:- Given a data set, one often assumes the data comes from an underlying
space. By imposing some discrete structure on the data, namely, a
simplicial complex, one can attempt to study the geometry of the space the
data is coming from by studying the geometry of the simplicial complex.
Classically, this has been accomplished via graphs (which are
1-dimensional simplicial complexes) through the graph Laplacian, which
approximates the Laplace-Beltrami operator on the underlying space.
However, higher-dimensional Laplacians exist which could give more
information. Recent applications of higher-dimensional Laplacians include
statistical ranking and parametrizing data via angular coordinates. In
this talk, some recent research on higher-dimensional Laplacians will be
discussed. Historically, graph Laplacians were fruitfully studied via
Cheeger numbers through the Cheeger inequality. A long-standing open
problem has been to extend those results to higher dimensions, a problem
which is partly addressed by our work. Finally, as time permits, the
behavior of higher-dimensional Laplacians (and Cheeger numbers) will be
illustrated via some examples, along with a possible future application of
higher-dimensional Laplacians.
Generated at 11:18am Saturday, April 20, 2024 by Mcal. Top
* Reload
* Login