On Lossy Compression of Generalized Gaussian Sources

This paper considers a problem of lossy compression of generalized Gaussian (GG) sources (i.e., sources with the probability density functions proportional to $\mathrm {e}^{-\frac {|x|^{\mathrm {S}}}{2}}$, $s \gt 0$) with an $\ell _{r}, r \gt 0$, distortion measure.It is shown that an optimal reconstruction distribution always exists and properties of this distribution are studied. In particular, it is shown that if $s \leq r-1$ then an optimal reconstruction must have unbounded support and for $s \gt r$ an optimal reconstruction must have bounded support. Further, it is shown that Shannon’s lower bound is achievable if and only if $r = s \in (0,1$] $\cup \{2\}$, or in other words when the GG distribution is self-decomposable. Finally, conditions are shown under which an optimal reconstruction is discrete with finitely many mass points.