Variable-Length Compression Allowing Errors

This paper studies the fundamental limits of the minimum average length of lossless and lossy variable-length compression, allowing a nonzero error probability ε, for lossless compression. We give nonasymptotic 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 - ε)kH(S) - ((kV(S)/2π))^1/2 exp[-((Q^-1 (ε))²/2)], where Q^-1 (·) is the functional inverse of the standard Gaussian complementary cumulative distribution function, and V(S) is the source dispersion. A nonzero error probability thus not only reduces the asymptotically achievable rate by a factor of 1 - ε, 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.