Abstract
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. From an information geometrical point of view, the evaluation of the rate-distortion function is achieved by a projection to the mixture family defined by the distortion measure. 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 at the minimum distortion level at zero rate. 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 power. These bounds evaluate from above the projection of the source distribution to the mixture family of the generalized Gaussian models. 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.
Highlights
The rate-distortion function, R(D), shows the minimum achievable rate to reproduce source outputs with the expected distortion not exceeding D
From Theorem 1, we immediately obtain the following corollary, which shows that the β-generalized Gaussian source is the only source that can make the Shannon lower bound (SLB) tight at D = Dmax under the β-th power distortion measure (14)
We have shown that the generalized Gaussian distribution is the only source that can make the SLB tight for all D under the power distortion measure if the orders of the source and the distortion measure are matched
Summary
The rate-distortion function, R(D), shows the minimum achievable rate to reproduce source outputs with the expected distortion not exceeding D. Using the bounds of the rate-distortion function of the β-th power difference distortion measure obtained in [11], we evaluate the projections of the source distribution to the mixture families associated with this distortion measure (Theorem 3). We prove that only the β-generalized Gaussian distribution has the potential to be the source whose SLB is tight; that is, identical to the rate-distortion function for the entire rage of positive rate This fact brings knowledge on the tightness of the SLB of an e-insensitive distortion measure, which is obtained by truncating the loss function near zero error [14,15,16]. The fact that the Laplacian (β = 1) and the Gaussian (β = 2) sources have the tight SLB derives a novel upper bound to R(D) of γ(6= β)-th power distortion measure, which has a constant gap from the SLB for all D. Extending the above argument to e-insensitive loss, we derive an upper bound to the distortion-rate function, which is tight in the limit of zero rate
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