Partially Linear Estimation with Application to Sparse Signal Recovery From Measurement Pairs

TL;DR

This paper introduces a partially linear MMSE estimator that relies only on second-order statistics, offering a computationally efficient alternative for sparse signal recovery and dynamic tracking with performance comparable to existing methods.

Contribution

The paper proposes a new PLMMSE estimator that does not require full distribution knowledge, providing a minimax-optimal, computationally efficient solution for signal recovery and tracking.

Findings

PLMMSE estimator performs close to state-of-the-art algorithms.

It offers significant computational speed advantages.

Effective in both static image enhancement and dynamic target tracking.

Abstract

We address the problem of estimating a random vector X from two sets of measurements Y and Z, such that the estimator is linear in Y. We show that the partially linear minimum mean squared error (PLMMSE) estimator does not require knowing the joint distribution of X and Y in full, but rather only its second-order moments. This renders it of potential interest in various applications. We further show that the PLMMSE method is minimax-optimal among all estimators that solely depend on the second-order statistics of X and Y. We demonstrate our approach in the context of recovering a signal, which is sparse in a unitary dictionary, from noisy observations of it and of a filtered version of it. We show that in this setting PLMMSE estimation has a clear computational advantage, while its performance is comparable to state-of-the-art algorithms. We apply our approach both in static and dynamic estimation applications. In the former category, we treat the problem of image enhancement from blurred/noisy image pairs, where we show that PLMMSE estimation performs only slightly worse than state-of-the art algorithms, while running an order of magnitude faster. In the dynamic setting, we provide a recursive implementation of the estimator and demonstrate its utility in the context of tracking maneuvering targets from position and acceleration measurements.