Constant-width rate-distortion bounds for power distortion measures

The explicit form of the rate-distortion function has rarely been obtained except for few cases where the Shannon lower bound coincides with the rate-distortion function for the entire range of the positive rate. In this paper, we consider the β-th power distortion measure, and prove that β-generalized Gaussian distribution is the only source that can make the Shannon lower bound tight for all distortion levels. We demonstrate that the tightness of the Shannon lower bound for β = 1 (Laplacian source) and β = 2 (Gaussian source) yields upper bounds to the rate-distortion function of power distortion measures with a different exponent, which have a constant gap from the Shannon lower bound for all distortion levels. Applying similar arguments to ε-insensitive distortion measures, we consider the tightness of the Shannon lower bound and derive an upper bound to the distortion-rate function, which is accurate at low rates.