Dimensionality Reduction for Distributed Estimation in the Infinite Dimensional Regime

Distributed estimation of an unknown signal is a common task in sensor networks. The scenario usually envisioned consists of several nodes, each making an observation correlated with the signal of interest. The acquired data is then wirelessly transmitted to a fusion center that aims at estimating the desired signal within a prescribed accuracy. Motivated by the obvious processing limitations inherent to such distributed infrastructures, we seek to find efficient compression schemes that account for limited available power and communication bandwidth. In this paper, we propose a transform-based approach to this problem where each sensor provides the fusion center with a low-dimensional approximation of its local observation by means of a suitable linear transform. Under the mean squared error criterion, we derive the optimal solution to apply at one sensor assuming all else being fixed. This naturally leads to an iterative algorithm whose optimality properties are exemplified using a simple though illustrative correlation model. The stationarity issue is also investigated. Under restrictive assumptions, we then provide an asymptotic distortion analysis, as the size of the observed vectors becomes large. Our derivation relies on a variation of the Toeplitz distribution theorem, which allows us to provide a reverse ldquowater-fillingrdquo perspective to the problem of optimal dimensionality reduction. We illustrate, with a first-order Gauss-Markov model, how our findings allow for the computation of analytical closed-form distortion formulas that provide an accurate estimation of the reconstruction error obtained in the finite-dimensional regime.