This paper presents an algorithm for a team of mobile robots to simultaneously learn a spatial field over a domain and spatially distribute themselves to optimally cover it. Drawing from previous approaches that estimate the spatial field through a centralized Gaussian process, this work leverages the spatial structure of the coverage problem and presents a decentralized strategy where samples are aggregated locally by establishing communications through the boundaries of a Voronoi partition. We present an algorithm whereby each robot runs a local Gaussian process calculated from its own measurements and those provided by its Voronoi neighbors, which are incorporated into the individual robot's Gaussian process only if they provide sufficiently novel information. The performance of the algorithm is evaluated in simulation and compared with centralized approaches.

In the panoply of pattern classification techniques, few enjoy the intuitive appeal and simplicity of the nearest neighbor rule: given a set of samples in some metric domain space whose value under some function is known, we estimate the function anywhere in the domain by giving the value of the nearest sample per the metric. More generally, one may use the modal value of the m nearest samples, where m is a fixed positive integer (although m=1 is known to be admissible in the sense that no larger value is asymptotically superior in terms of prediction error). The nearest neighbor rule is nonparametric and extremely general, requiring in principle only that the domain be a metric space. The classic paper on the technique, proving convergence under independent, identically-distributed (iid) sampling, is due to Cover and Hart (1967). Because taking samples is costly, there has been much research in recent years on selective sampling, in which each sample is selected from a pool of candidates ranked by a heuristic; the heuristic tries to guess which candidate would be the most "informative" sample. Lindenbaum et al. (2004) apply selective sampling to the nearest neighbor rule, but their approach sacrifices the austere generality of Cover and Hart; furthermore, their heuristic algorithm is complex and computationally expensive. Here we report recent results that enable selective sampling in the original Cover-Hart setting. Our results pose three selection heuristics and prove that their nearest neighbor rule predictions converge to the true pattern. Two of the algorithms are computationally cheap, with complexity growing linearly in the number of samples. We believe that these results constitute an important advance in the art.