Second-Order Asymptotics of Rate-Distortion using Gaussian Codebooks for Arbitrary Sources

The rate-distortion saddle-point problem considered by Lapidoth (1997) consists in finding the minimum rate to compress an arbitrary ergodic source when one is constrained to use a random Gaussian codebook and minimum (Euclidean) distance encoding is employed. We extend Lapidoth's analysis in several directions in this paper. Firstly, we consider second-order asymptotics. In particular, when the source is stationary and memoryless, we establish ensemble tight second-order coding rate for the problem. Secondly, by 'random Gaussian codebook', Lapidoth refers to a collection of random codewords, each of which is drawn independently and uniformly from the surface of an n-dimensional sphere. To be more precise, we term this as a spherical Gaussian codebook. We also consider i.i.d. Gaussian codebooks in which each random codeword is drawn independently from a product Gaussian distribution. We also derive the second-order asymptotics when i.i.d. Gaussian codebooks are employed. Interestingly, in contrast to the recent work on the channel coding counterpart by Scarlett, Tan and Durisi (2017), the dispersions for spherical and i.i.d. Gaussian code books are identical for the rate-distortion saddle-point problem.