A Lower Bound on the Estimator Variance for the Sparse Linear Model

TL;DR

This paper derives a new lower bound on the variance of estimators for sparse vectors in noisy linear models, improving existing bounds especially in the case of unbiased estimation without linear transformation.

Contribution

It introduces a novel lower bound on estimator variance using RKHS framework applicable to general linear transformations and bias functions, including compressed sensing scenarios.

Findings

New lower bound on estimator variance derived using RKHS.

Bound improves upon previous bounds for unbiased sparse vector estimation.

Applicable to underdetermined systems and general bias functions.

Abstract

We study the performance of estimators of a sparse nonrandom vector based on an observation which is linearly transformed and corrupted by additive white Gaussian noise. Using the reproducing kernel Hilbert space framework, we derive a new lower bound on the estimator variance for a given differentiable bias function (including the unbiased case) and an almost arbitrary transformation matrix (including the underdetermined case considered in compressed sensing theory). For the special case of a sparse vector corrupted by white Gaussian noise-i.e., without a linear transformation-and unbiased estimation, our lower bound improves on previously proposed bounds.