Variable-length compression allowing errors

Abstract

This paper studies the fundamental limits of the minimum average length of lossless and lossy variable-length compression, allowing a nonzero error probability $\epsilon$, for lossless compression. We give non-asymptotic bounds on the minimum average length in terms of Erokhin's rate-distortion function and we use those bounds to obtain a Gaussian approximation on the speed of approach to the limit which is quite accurate for all but small blocklengths: $$(1-\epsilon )kH(\mathsf{S})-\sqrt{\frac{kV(\mathsf{S})}{2\pi }}{e}^{-\frac{(Q^{-1}(\epsilon ){)}^2}{2}}$$ where $Q^{-1}(\cdot )$ is the functional inverse of the standard Gaussian complementary cdf, and $V(\mathsf{S})$ is the source dispersion. A nonzero error probability thus not only reduces the asymptotically achievable rate by a factor of $1-\epsilon$, but this asymptotic limit is approached from below, i.e. larger source dispersions and shorter blocklengths are beneficial. Variable-length lossy compression under an excess distortion constraint is shown to exhibit similar properties.