Discrete Optimal Reconstruction Distributions for Itakura-Saito Distortion Measure

The optimal reconstruction distribution achieving the rate-distortion function is elusive except for limited examples of sources and distortion measures if the rate-distortion function is strictly greater than the Shannon lower bound. In this paper, focusing on the Itakura-Saito distortion measure, we prove that if the Shannon lower bound is not tight, the optimal reconstruction distribution is purely discrete. Combined with the fact that the Shannon lower bound is tight for the gamma source, this result shows that it is the only source that has continuous optimal reconstruction distributions for the range of entire positive rate.