Limits to estimation in stochastic ill-conditioned inverse problems

Abstract

Using information-theoretic methods we develop simple results quantifying a lower bound for minimax estimation, a kind of infinite-dimensional Cramer-Rao lower bound, for signal estimation in possibly nonlinear, ill-conditioned, inverse problems. Our results reduce calculation to a geometric computation based on a modulus of continuity and make explicit connections with results in the literature on deterministic ill-conditioned inverse problems. Several applications are discussed.