# Sparse-Based Estimation Performance for Partially Known Overcomplete Large-Systems

## TL;DR

This paper develops a low-cost, accurate Bayesian lower bound for estimating sparse signals in large systems with known interfering subspaces, combining compressed sensing and random matrix theory.

## Contribution

It introduces a novel closed-form Bayesian lower bound for sparse vector estimation that leverages CS and RMT, applicable in doubly asymptotic regimes.

## Key findings

- Closed-form lower bounds accurately predict MSE of sparse estimators.
- The bounds are computationally efficient and interpretable.
- Joint estimation/rejection improves signal recovery in noisy scenarios.

## Abstract

We assume the direct sum <A> o <B> for the signal subspace. As a result of post- measurement, a number of operational contexts presuppose the a priori knowledge of the LB -dimensional "interfering" subspace <B> and the goal is to estimate the LA am- plitudes corresponding to subspace <A>. Taking into account the knowledge of the orthogonal "interfering" subspace <B>\perp, the Bayesian estimation lower bound is de- rivedfortheLA-sparsevectorinthedoublyasymptoticscenario,i.e. N,LA,LB -> \infty with a finite asymptotic ratio. By jointly exploiting the Compressed Sensing (CS) and the Random Matrix Theory (RMT) frameworks, closed-form expressions for the lower bound on the estimation of the non-zero entries of a sparse vector of interest are derived and studied. The derived closed-form expressions enjoy several interesting features: (i) a simple interpretable expression, (ii) a very low computational cost especially in the doubly asymptotic scenario, (iii) an accurate prediction of the mean-square-error (MSE) of popular sparse-based estimators and (iv) the lower bound remains true for any amplitudes vector priors. Finally, several idealized scenarios are compared to the derived bound for a common output signal-to-noise-ratio (SNR) which shows the in- terest of the joint estimation/rejection methodology derived herein.