|Ph.D Student||Ben-Haim Zvi|
|Subject||Performance Bounds for Estimation of Structurally|
|Department||Department of Electrical Engineering||Supervisor||Professor Yonina Eldar|
|Full Thesis text|
The goal of this work is to analyze the performance of various signal estimation techniques in both the Bayesian and structured frequentist settings. The theoretical foundations of our work are based on several extensions on the Cramer-Rao bound (CRB). Specifically, in the frequentist setting, our results rely on a novel interpretation of the effect of parametric constraints on the CRB, as well as a generalization of the CRB for estimating continuous-time functions. While the CRB is a bound for the frequentist (or deterministic) scenario, we show that with an appropriate adaptation, the bound can be applied in the Bayesian world as well, and derive a general lower bound on the minimum achievable MSE in Bayesian estimation settings.
The aforementioned results are complemented by upper bounds on the performance of specific estimators. Such upper bounds can be thought of as guarantees which ensure that a specific approach will never be worse than an analytically computable limit. The combination of upper and lower bounds identifies cases in which state-of-the-art techniques approach the theoretically optimal performance, while illuminating scenarios in which existing methods can be improved.
We demonstrate the applicability of our work in three practical estimation problems: sparsely representable signals, block-sparse parameters, and the finite rate of innovation (FRI) scenario. We demonstrate that under appropriate assumptions, sparse and block-sparse estimation algorithms come fairly close to the theoretical limit. The situation is less favorable in the case of FRI estimation, where there is often a large gap between existing techniques and the lower bound, implying that there is still room for improvement of FRI methods. This analysis allows us to distinguish fundamental limitations in the problem setting from flaws in existing algorithms, thus pinpointing directions for the development of improved estimation techniques.