The Uncertain Generalized Moment Problem with Complexity Constraint

Abstract

This paper is dedicated to Arthur Krener – a great researcher, a great teacher and a great friend – on the occasion of his 60th birthday. In this work we study the generalized moment problem with complexity constraints in the case where the actual values of the moments are uncertain. For example, in spectral estimation the moments correspond to estimates of covariance lags computed from a finite observation record, which inevitably leads to statistical errors, a problem studied earlier by Shankwitz and Georgiou. Our approach is a combination of methods drawn from optimization and the differentiable approach to geometry and topology. In particular, we give an intrinsic geometric derivation of the Legendre transform and use it to describe convexity properties of the solution to the generalized moment problems as the moments vary over an arbitrary compact convex set of possible values. This is also interpreted in terms of minimizing the Kullback-Leibler divergence for the generalized moment problem.

This research was supported in part by grants from AFOSR, VR, Institut Mittag- Leffler, and SBC.