Fixed-length lossy compression in the finite blocklength regime

TL;DR

This paper derives tight bounds for lossy source coding rates at finite blocklengths, showing they closely approximate a normal approximation involving rate-distortion and dispersion for stationary memoryless sources.

Contribution

It provides the first finite blocklength bounds for lossy compression, explicitly characterizing the minimum rate with a normal approximation involving dispersion.

Findings

Achievability and converse bounds valid at any fixed blocklength.

Approximate minimum rate closely matches $R(d) + rac{ ext{sqrt}(V(d)/n)}{Q^{-1}( ext{epsilon})}$.

Results apply to stationary memoryless sources with separable distortion.

Abstract

This paper studies the minimum achievable source coding rate as a function of blocklength $n$ and probability $\epsilon$ that the distortion exceeds a given level $d$. Tight general achievability and converse bounds are derived that hold at arbitrary fixed blocklength. For stationary memoryless sources with separable distortion, the minimum rate achievable is shown to be closely approximated by $R(d) + \sqrt{\frac{V(d)}{n}} Q^{-1}(\epsilon)$, where $R(d)$ is the rate-distortion function, $V(d)$ is the rate dispersion, a characteristic of the source which measures its stochastic variability, and $Q^{-1}(\epsilon)$ is the inverse of the standard Gaussian complementary cdf.