Variable-Length Compression Allowing Errors

This paper studies the fundamental limits of the minimum average length of variable-length compression when a nonzero error probability ε is tolerated. 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-ε)kH(S) - √kV(S)/2π e – (Q^(-1)(ε))^2/2 where Q-1 (·) is the functional inverse of the Q-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 also this asymptotic limit is approached from below, i.e. a larger source dispersion and shorter blocklengths are beneficial. Further, we show that variable-length lossy compression under excess distortion constraint also exhibits similar properties.