  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.
