Towards Empirical Sandwich Bounds on the Rate-Distortion Function

TL;DR

This paper introduces an empirical method to estimate sandwich bounds on the rate-distortion function for general data sources using only i.i.d. samples, providing insights into the limits of data compression.

Contribution

It presents the first algorithm to empirically sandwich the R-D function for non-discrete sources with only i.i.d. data samples, extending beyond previous distributional assumptions.

Findings

Estimated R-D bounds for artificial and real-world data.

Indicated potential for improving image compression by at least 1 dB PSNR.

Provided open-source code and data for reproducibility.

Abstract

Rate-distortion (R-D) function, a key quantity in information theory, characterizes the fundamental limit of how much a data source can be compressed subject to a fidelity criterion, by any compression algorithm. As researchers push for ever-improving compression performance, establishing the R-D function of a given data source is not only of scientific interest, but also sheds light on the possible room for improving compression algorithms. Previous work on this problem relied on distributional assumptions on the data source (Gibson, 2017) or only applied to discrete data (Blahut, 1972; Arimoto, 1972). By contrast, this paper makes the first attempt at an algorithm for sandwiching the R-D function of a general (not necessarily discrete) source requiring only i.i.d. data samples. We estimate R-D sandwich bounds for a variety of artificial and real-world data sources, in settings far beyond the feasibility of any known method, and shed light on the optimality of neural data compression (Ball\'e et al., 2021; Yang et al., 2022). Our R-D upper bound on natural images indicates theoretical room for improving state-of-the-art image compression methods by at least one dB in PSNR at various bitrates. Our data and code can be found at https://github.com/mandt-lab/empirical-RD-sandwich.