Fall 2005 Seminar Series

DATE: Thursday, December 8, 2005
LOCATION: E40-298
TIME: 4:15pm
Reception immediately following in the Philip M. Morse Reading Room, E40-106

SPEAKER:
Professor Stephen P. Boyd
Samsung Professor of Engineering and
Professor of Electrical Engineering in the Information Systems Laboratory
Stanford University

TITLE
A New Minimax Theorem for Rayleigh Quotients, with Applications to Machine Learning, Signal Processing, and Finance

ABSTRACT
We consider a function of the form $(a^Tx)^2/x^TBx$, where $B$ is positive definite. This is a special type of Rayleigh quotient, and arises in several applications. We consider the game of choosing $x$ from a convex set, to maximize the function, and choosing $(a,B)$ from a convex set, to minimize it. We give a nonstandard minimax theorem for this game, and describe an efficient method for computing the saddle point, based on convex optimization.

 

We describe applications in machine learning (robust Fisher linear discriminant analysis), signal processing (robust beamforming, robust matched filtering), and finance (maximizing the worst-case Sharpe ratio). In these applications, $x$ corresponds to some design or model parameters to be chosen, and the pair $(a,B)$ corresponds to uncertainty in the statistical model.

 

Joint work with Seung Jean Kim and Alessandro Magnani