Distributed multi-agent Gaussian regression via Karhunen-Loève expansions

We consider the problem of distributedly estimating Gaussian random fields in multi-agent frameworks. Each sensor collects few measurements and aims to collaboratively reconstruct a common estimate based on all data. Agents are assumed to have limited computational and communication capabilities and to gather M noisy measurements in total on input locations independently drawn from a known common probability density. The optimal solution would require agents to exchange all the M input locations and measurements and then invert an M ×M matrix, a non-scalable task. Differently, we propose two suboptimal approaches using the first E orthonormal eigenfunctions obtained from the Karhunen-Loève (KL) expansion of the chosen kernel, where typically E M . The benefit is twofold: first, the computation and communication complexities scale with E and not with M . Second, computing the required sufficient statistics can be performed via standard average consensus algorithms. We obtain probabilistic non-asymptotic bounds for both approaches, useful to determine a priori the desired level of estimation accuracy. Furthermore, we also derive new distributed strategies to tune the regularization parameter which rely on the Stein’s unbiased risk estimate (SURE) paradigm and can again be implemented via standard average consensus algorithms. The proposed estimators and bounds are finally tested on both synthetic and real field data. Index Terms Gaussian processes, sensor networks, distributed estimation, kernel-based regularization, nonparametric estimation, average consensus