We study the compression of data in the case where the useful information is contained in a set rather than a vector, i.e., the ordering of the data points is irrelevant and the number of data points is unknown. Our analysis is based on rate-distortion theory and the theory of finite point processes. We introduce fundamental information-theoretic concepts and quantities for point processes and present general lower and upper bounds on the rate-distortion function. To enable a comparison with the vector setting, we concretize our bounds for point processes of fixed cardinality. In particular, we analyze a fixed number of unordered Gaussian data points and show that we can significantly reduce the required rates compared to the best possible compression strategy for Gaussian vectors. As an example of point processes with variable cardinality, we study the best possible compression of Poisson point processes. For the specific case of a Poisson point process with uniform intensity on the unit square, our lower and upper bounds are separated by only a small gap and thus provide a good characterization of the rate-distortion function.

We consider the point process of signal strengths emitted from transmitters in a wireless network and observed at a fixed position. In our model, transmitters are placed deterministically or randomly according to a hard core or Poisson point process, and the signals are subjected to power law propagation loss and random propagation effects that may be correlated between transmitters. We provide bounds on the distance between the point process of signal strengths and a Poisson process with the same mean measure, assuming correlated log-normal shadowing. For “strong shadowing” and moderate correlations, we find that the signal strengths are close to a Poisson process, generalizing a recently shown analogous result for independent shadowing.