Fast Relative Entropy Coding with A* coding

TL;DR: A* coding with (IK)VAEs on MNIST is evaluated, showing that it can losslessly compress images near the theoretically optimal limit and the IsoKL VAE, which can be used with DAD* to further improve compression efficiency, is proposed.

Abstract: Relative entropy coding (REC) algorithms encode a sample from a target distribution Q using a proposal distribution P , such that the expected codelength is O ( D KL [ Q ∥ P ]) . REC can be seam-lessly integrated with existing learned compression models since, unlike entropy coding, it does not assume discrete Q or P , and does not require quantisation. However, general REC algorithms require an intractable Ω( e D KL [ Q ∥ P ] ) runtime. We introduce AS* and AD* coding, two REC algorithms based on A* sampling. We prove that, for continuous distributions over R , if the density ratio is unimodal, AS* has O ( D ∞ [ Q ∥ P ]) expected runtime, where D ∞ [ Q ∥ P ] is the R´enyi ∞ divergence. We provide experimental evidence that AD* also has O ( D ∞ [ Q ∥ P ]) expected runtime. We prove that AS* and AD* achieve an expected codelength of O ( D KL [ Q ∥ P ]) . Further, we introduce DAD*, an approximate algorithm based on AD* which retains its favourable runtime and has bias similar to that of alternative methods. Focusing on VAEs, we propose the IsoKL VAE (IKVAE), which can be used with DAD* to further improve compression efficiency. We evaluate A* coding with (IK)VAEs on MNIST, showing that it can losslessly compress images near the theoretically optimal limit.