Theoretical Bounds on MAP Estimation in Distributed Sensing Networks

TL;DR

This paper provides theoretical bounds on MAP estimation in distributed sensing networks, demonstrating a decoupling property and proposing a novel multi-dimensional soft thresholding algorithm that outperforms traditional methods.

Contribution

It introduces a theoretical analysis of MAP estimation in distributed networks and proposes a new algorithm for jointly sparse signals with improved performance.

Findings

Decoupling property of the setting shown via the replica method.

Proposed multi-dimensional soft thresholding algorithm outperforms $\, ext{l}_{2,1}$-norm regularized least squares.

Algorithm achieves better recovery with feasible computational complexity.

Abstract

The typical approach for recovery of spatially correlated signals is regularized least squares with a coupled regularization term. In the Bayesian framework, this algorithm is seen as a maximum-a-posterior estimator whose postulated prior is proportional to the regularization term. In this paper, we study distributed sensing networks in which a set of spatially correlated signals are measured individually at separate terminals, but recovered jointly via a generic maximum-a-posterior estimator. Using the replica method, it is shown that the setting exhibits the decoupling property. For the case with jointly sparse signals, we invoke Bayesian inference and propose the "multi-dimensional soft thresholding" algorithm which is posed as a linear programming. Our investigations depict that the proposed algorithm outperforms the conventional $\ell_{2,1}$-norm regularized least squares scheme while enjoying a feasible computational complexity.