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We consider a random geometric graph with vertices sampled from a probabilitymeasure supported on $\mathbb R^d$, and study its connectivity. We show thegraph is typically disconnected, unless the sampling density hassuperexponential decay. In the later setting, we identify an asymptoticthreshold value for the radius parameter of the graph such that, for radiusvalues beyond the threshold, some concentration properties hold for the sampledpoints of the graph, while the graph is disconnected for radius values belowthe same threshold. Properties of point processes are well-known to be closelyrelated to the analysis of geometric learning problems, such as spectralclustering. This work can be seen as a first step towards understanding theconsistency of spectral clustering when the probability measure has unboundedsupport. In particular, we narrow down the setting under which spectralclustering algorithms on $\mathbb R^d$ may be expected to achieve consistency,to a sufficiently fast decay of the sampling density (superexponential) and asufficiently slowly decaying radius parameter value as a function of $n$, thenumber of sampled points.

Which Sampling Densities are Suitable for Spectral Clustering on Unbounded Domains?