2006-07 Program on High Dimensional Inference and Random Matrices

Second Semester Emphasis on Geometry and Random Matrices

Research Foci
Description of Activities

• Workshop on Geometry, Random Matrices and Statistical Inference, Jan. 16-19, 2007
• Course on Geometry, Random Matrices, and Statistical Inference

Research Foci

From the perspective of inference, clustering, and machine learning, geometric ideas have been gaining greater emphasis. One reason for this has been the realization that predictive models with a small amount of labeled data can be greatly improved by incorporating unlabelled data. Thus the geometry of the marginal distribution provides salient and compelling information in many real world problems.

This insight has lead to a variety of statistical models and algorithms as well as to the study of a variety of mathematical objects. A nonexhaustive list follows: spectral clustering, nonlinear dimensionality reduction, manifold learning, learning homologies, topological persistence, semi-supervised learning, non-parametric semi-supervised Bayesian models, the Laplace-Beltrami operator, graph Laplacians, diffusion models on manifolds, random projections. Most, if not all, of the above topics are intimately related to the study of random matrices either from an algorithmic perspective or from the perspective that the structure of a random matrix depending on data drawn from a measure is fundamental in understanding the topic.

There is interest in this area both from statisticians involved in statistical inference of high-dimensional data, computer scientists interested in computational geometry and topology, and mathematicians studying multiscale representations and diffusion processes.

Description of Activities

The week long workshop will begin the program focusing on Geometry and Random Matrices. Both algorithms and the fundamental mathematical objects computed by the algorithms will be stressed. This will be followed by a semester long working group on "Geometry, Random Matrices, and Statistical Inference." It will be a part of the High Dimensional Inference and Random Matrices Program at SAMSI.

This semester emphasis will bring together computer scientists, geometers, and statisticians to explore geometric ideas in statistical inference and computational methods for inference. High dimensional inference and random matrices are central quantities and structures that need to be studied and understood for this program.

We are also planning to have regular visitors, who will give talks at the weekly seminar. Some of the visitors may stay for longer periods of time, e.g. 1-2 weeks. There will be a SAMSI course (Geometry, Random Matrices, and Statistical Inference) associated with the program.

Committees:

• Organizing Committtee: Misha Belkin (The Ohio State University, Chair), Feng Liang (University of Illinois at Urbana-Champaign)
• Local Scientific Coordinator: Sayan Mukherjee (Duke University)
• Scientific Committee: Mauro Maggioni (Duke University), Yury Mileyko (Duke University), Herbert Edelsbrunner, (Duke University)

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