Bayesian Lower Bounds for Dense or Sparse (Outlier) Noise in the RMT Framework

TL;DR

This paper derives Bayesian lower bounds on estimation error for models with Student's t-distributed noise, applicable to both dense and sparse noise scenarios, using random matrix theory for asymptotic analysis.

Contribution

It introduces a novel framework combining Bayesian bounds and RMT to compare estimation performance under different noise distributions, including outliers.

Findings

Closed-form BCRB expressions for Student's t noise

Comparison framework for dense vs. sparse noise models

Asymptotic analysis with RMT techniques

Abstract

Robust estimation is an important and timely research subject. In this paper, we investigate performance lower bounds on the mean-square-error (MSE) of any estimator for the Bayesian linear model, corrupted by a noise distributed according to an i.i.d. Student's t-distribution. This class of prior parametrized by its degree of freedom is relevant to modelize either dense or sparse (accounting for outliers) noise. Using the hierarchical Normal-Gamma representation of the Student's t-distribution, the Van Trees' Bayesian Cram\'er-Rao bound (BCRB) on the amplitude parameters is derived. Furthermore, the random matrix theory (RMT) framework is assumed, i.e., the number of measurements and the number of unknown parameters grow jointly to infinity with an asymptotic finite ratio. Using some powerful results from the RMT, closed-form expressions of the BCRB are derived and studied. Finally, we propose a framework to fairly compare two models corrupted by noises with different degrees of freedom for a fixed common target signal-to-noise ratio (SNR). In particular, we focus our effort on the comparison of the BCRBs associated with two models corrupted by a sparse noise promoting outliers and a dense (Gaussian) noise, respectively.