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2006-07 Program on High Dimensional Inference and Random Matrices
Research Foci
Description of Activities
- Opening Workshop, Sept. 17-20, 2006
- Working Groups
- Random Matrices Course
- Bayesian Focus Week, Oct.30 - Nov. 3, 2006
- Large Graphical Models and Random Matrices, Nov. 9-11, 2006
- Joint NCAR and SAMSI workshop: Geophysical Models at NCAR: A Scoping and
Synthesis Workshop, Nov. 13-14, 2006
- Workshop on Geometry, Random Matrices and Statistical Inference, Jan. 16-19, 2007
- Course on Geometry, Random Matrices, and Statistical Inference
- Second Semester Emphasis on Geometry and Random Matrices
- Transition Workshop, April 10-13, 2007 at AIM
- Application of Random Matrices: Theory and Methods, May 7-9 at NCAR
Further Information
RANDOM MATRIX THEORY lies at the confluence of several areas of mathematics, especially number
theory, combinatorics, dynamical systems, diffusion processes, probability and statistics. At the
same time Random Matrix Theory may hold the key to solving critical problems for a broad range of
complex systems from biophysics to quantum chaos to signals and communication theory to machine
learning to finance to geoscience modeling. This semester-long Program is a unique opportunity to
explore the interplay of stochastic and mathematical aspects to random matrix theory and application.
Program Leaders: Iain Johnstone (Stanford University, Chair), Peter Bickel (UC Berkeley), Helene Massam (York University),
Douglas Nychka (NCAR), Craig Tracy (UC Davis); G. W. Stewart (Univ. of Maryland, National Advisory Committee Liaison),
Chris Jones (SAMSI, Directorate Liaison)
Scientific Committee: Myles Allen (Oxford), Estelle Basor (California Polytechnic, San Luis
Obispo), David Donoho (Statistics, Stanford), Persi Diaconis (Statistics, Stanford), Jianqing Fan
(Princeton), Ken McLaughlin (Mathematics, Univ. of Arizona), Neil O'Connell (Univ. of Warwick, UK),
Ben Santner (Lawrence Livermore), Jack Silverstein (Mathematics, N.C. State), Ofer Zeitouni
(Univ. of Minnesota)
Research Foci
The aim of the Program is to bring together researchers interested in the theory and applications
of random matrices to share their results, discuss new research directions and develop
collaborations. The focus of the Program will be on large-dimensional random matrices and the
problems which make use of them.
At least three broad challenges stand out:
- Furthering our understanding of the spectral properties of random matrices under various
models and assumptions (this is a "direct" problem).
- Recovering from an observed random matrix information about the process from which it
was generated (this is an "inverse" problem).
- Making an efficient use of newly gained understanding of random matrices to advance
research in a wide array of scientific disciplines, including statistics, dynamical systems,
climatology, machine learning, signal processing, and finance.
Although specific research foci will be determined by the participants following the Opening
Tutorial and Kickoff Workshop, potential research topics might draw from:
Direct Problems:
- Extreme eigenvalues of random covariance matrices:
Asymptotic and non-asymptotic distributions, in the Gaussian setting. Robustness of
results to Gaussian assumption. Focus on non-diagonal covariance matrices. Difficulties arising
with real-valued random variables.
- Dynamic behavior of eigenvalues of matrix processes:
Matrices whose elements undergo diffusion (Dyson Processes); stochastic differential
equations for their eigenvalues. Scaling limits (Airy processes) and their descriptions via
Partial Differential Equations. Connections to growth processes.
- Eigenvector problems:
Limiting theory for eigenvectors. Free probability techniques.
- Spectral properties of other random matrices arising in multivariate Statistics:
Techniques such as Canonical Correlation Analysis and related random matrix
problems. Covariance matrices with covariance in space and time.
Inverse problems:
- Estimation of large covariance matrices:
Regularization techniques by banding, filtering and using L1 penalties; their theoretical
and practical properties. Problems of space-time models. Study of methods used in scientific
disciplines such as machine learning and "econophysics".
- Consistency and estimability problems:
Consistency of sample eigenvectors in covariance estimation problems. Estimability
issues for eigenvectors and eigenvalues. Implications for Principal Component Analysis (PCA).
Applications:
- Climatology:
Empirical Orthogonal Functions and related techniques: summarization of evolving
geophysical fields via spectral techniques. Approximation via truncated expansions, properties of
these truncated expansions (feature significance). Regularization techniques including tapering
and inflation methods. Application to detection and estimation of climate change signal.
- Dynamical systems:
Design of snapshots, i.e., finding a minimal number of functions, tailored to a
specific problem, that accurately represent the problem's dynamics.
- Data Assimilation:
Combination of numerical models and observations. Ensemble Kalman filtering: impact
of sample information on propagation of covariance, effect of ensemble size, tapering and
inflation methods.
- Graphical Gaussian models:
Consistent estimation of the model structure,study of the rates of convergence.
Consistent estimation of the covariance matrix. Other problems arising in graphical models with
large data sets.
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Computation of moments of large random Wishart matrices, connection to graph theory
and connection to methods in free probability.
- General statistical inference:
Consistency of regression functions dependent upon the behaviour of certain large
random matrices. Problems arising in model selection, estimation and testing. Principal components
analysis and choice of the best projections.
Description of Activities
Workshops: The Kickoff Workshop will be September 17, 2006 - September 20, 2006 will include
a one-day Opening Tutorial to present background on random matrices viewed from each key mathematical
discipline. The principal goal of the Workshop itself will be to engage a broad mathematical base
to focus on open questions that are amenable to solution by combining probabilistic and applied
mathematical approaches.
In Spring 2007, the Research Transition Workshop will provide a forum for presentation of results
and applications.
Working Groups: The working groups meet regularly throughout the program to pursue particular
research topics identified in the kickoff workshop (or subsequently chosen by the working group
participants). The working groups consist of SAMSI visitors, postdoctoral fellows, graduate students,
and local faculty and scientists. It is not necessary to be continually resident at SAMSI to
maintain connection to the working groups.
Climate and Weather
Wireless Communications
Universality
Regularization and Covariance
Geometric Methods
Multivariate Distributions
Graphical Models/Bayesian Methods
Estimating functionals of a high dimensional sparse vector of means
Further Information
Additional information about the program and opportunities to participate in it is available:
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