Rate-Distortion Problems of the Poisson Process: a Group-Theoretic Approach

We study rate-distortion problems of a Poisson process using a group theoretic approach. By describing a realization of a Poisson point process with either point timings or inter-point intervals and by choosing appropriate distortion measures, we establish rate-distortion problems of a homogeneous Poisson process as ball-or sphere-covering problems for realizations of the hyperoctahedral group in $\mathbb{R}^{n}$. Specifically, the realizations we investigate are a hypercube and a hyperoctahedron. Thereby we unify three known rate-distortion problems of a Poisson process (with different distortion measures, but resulting in the same rate-distortion function) with the Laplacian-$\ell_{1}$ rate-distortion problem.