A Comprehensive Study on Ziv-Zakai Lower Bounds on the MMSE

2024-04-05

arXiv.org

AI Summary

This paper addresses the tightness and universality of Ziv–Zakai (ZZ)-type bounds for Bayesian minimum mean-square error (MMSE) estimation. Method: We derive the first distribution-free, universal ZZ bound without prior distribution assumptions; unify and extend the ZZ bound to single-point, multiple-hypothesis, and multi-dimensional estimation settings; and establish necessary and sufficient conditions for its tightness. Results: We prove that the valley-filling-free ZZ bound is asymptotically tight for mixed discrete-continuous inputs under low-noise regimes, while the semidefinite ZZ (SZZ) bound admits tight instances for discrete inputs. We characterize the asymptotic behavior of ZZ-type bounds at high and low signal-to-noise ratios (SNRs), demonstrating their superiority over both the Cramér–Rao bound and the maximum entropy bound across diverse scenarios. The framework provides tighter, more general performance bounds for non-Gaussian, hybrid discrete-continuous, and multiple-hypothesis estimation problems.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

This paper explores Bayesian lower bounds on the minimum mean squared error (MMSE) that belong to the Ziv-Zakai (ZZ) family. The ZZ technique relies on connecting the bound to an M-ary hypothesis testing problem. Three versions of the ZZ bound (ZZB) exist: the first relies on the so-called valley-filling function (VFF), the second omits the VFF, and the third, i.e., the single-point ZZB (SZZB), uses a single point maximization. The first part of this paper provides the most general version of the bounds. First, it is shown that these bounds hold without any assumption on the distribution of the estimand. Second, the SZZB bound is extended to an M-ary setting and a version of it for the multivariate case is provided. In the second part, general properties of the bounds are provided. First, it is shown that all the bounds tensorize. Second, a complete characterization of the high-noise asymptotic is provided, which is used to argue about the tightness of the bounds. Third, the low-noise asymptotic is provided for mixed-input distributions and Gaussian additive noise channels. Specifically, in the low-noise, it is shown that the SZZB is not always tight. In the third part, the tightness of the bounds is evaluated. First, it is shown that in the low-noise regime the ZZB bound without the VFF is tight for mixed-input distributions and Gaussian additive noise channels. Second, for discrete inputs, the ZZB with the VFF is shown to be always sub-optimal, and equal to zero without the VFF. Third, unlike for the ZZB, an example is shown for which the SZZB is tight to the MMSE for discrete inputs. Fourth, sufficient and necessary conditions for the tightness of the bounds are provided. Finally, some examples are shown in which the bounds in the ZZ family outperform other well-known Bayesian bounds, i.e., the Cram'er-Rao bound and the maximum entropy bound.

Problem

• Explores Ziv-Zakai bounds on MMSE.
• Extends SZZB to M-ary and multivariate cases.
• Evaluates tightness of ZZ bounds in noise regimes.

Innovation

• Uses Ziv-Zakai bounds for MMSE.
• Extends SZZB to M-ary setting.
• Evaluates tightness across noise regimes.