We introduce a new distortion measure for point processes called functional-covering distortion. It is inspired by intensity theory and is related to both the covering of point processes and logarithmicloss distortion. We obtain the distortion-rate function with feedforward under this distortion measure for a large class of point processes. For Poisson processes, the rate-distortion function is obtained under a general condition called constrained functional-covering distortion, of which both covering and functional-covering are special cases. Also for Poisson processes, we characterize the rate-distortion region for a two-encoder CEO problem and show that feedforward does not enlarge this region.
I. INTRODUCTIONThe classical theory of compression [2] focuses on discrete-time, sequential sources. The theory is thus well-suited to text, audio, speech, genomic data, and the like. Continuous-time signals are typically handled by reducing to discrete-time via projection onto a countable basis. Multi-dimensional extensions enable application to images and video. Point processes model a distinct data type that appears in diverse domains such as neuroscience [3]-[8], communication networks [9]-[11], imaging [12], [13], blockchains [14]-[17],and photonics [18]- [22]. Formally, a point process can be viewed as a random counting measure on some space of interest [23], or if the space is a real line, a random counting function; we shall adopt the latter view. Informally, it may be viewed as simply a random collection of points representing epochs in time or points in space.