Given a data set, one often assumes the data comes from an underlying space. By imposing some discrete structure on the data, like a simplicial complex, one can attempt to study the geometry of the space that 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, and the possibility of applying higher-dimensional Laplacians to the problem of dimension reduction will be discussed.
A Cheeger-Type Inequality on Simplicial Complexes
John Steenbergen, Department of Mathematics, Duke University
Nov 21 2012 - 3:45pm
102A McDonnell Hall