Pragmatic lossless compression: Fundamental limits and universality
Andreas Theocharous, Lampros Gavalakis, Ioannis Kontoyiannis
TL;DR
The paper analyzes variable-rate, lossless compression of memoryless sources in regimes where the excess-rate probability decays exponentially with blocklength, deriving sharp nonasymptotic bounds that reveal a pragmatic rate ${\cal R}_n(\epsilon,P)$ governed by the inverse error-exponent function rather than the entropy. It provides both achievability and converse results for known sources, and proves a universal achievability for all sources on a finite alphabet, incurring a universality penalty that scales as $\left(\frac{m-2}{2}-\frac{1}{2(1-\alpha^*)}\right)\log n$. A central technical contribution is a precise count of low-entropy types, enabling the universal results, with the key bound $|\{x^n: H(\hat{P}_{x^n})\le H(Q)\}|=\Theta(n^{(m-3)/2}2^{nH(Q)})$. Overall, the work shows that for short blocklengths and stringent rate guarantees, the best achievable rate is substantially larger than the entropy and well-approximated by the finite-n pragmatic rate, while universal one-to-one coding can achieve near-optimal rates at a known alphabet-size penalty.
Abstract
The problem of variable-rate lossless data compression is considered, for codes with and without prefix constraints. Sharp bounds are derived for the best achievable compression rate of memoryless sources, when the excess-rate probability is required to be exponentially small in the blocklength. Accurate nonasymptotic expansions with explicit constants are obtained for the optimal rate, using tools from large deviations and Gaussian approximation. When the source distribution is unknown, a universal achievability result is obtained with an explicit ''price for universality'' term. This is based on a fine combinatorial estimate on the number of sequences with small empirical entropy, which might be of independent interest. Examples are shown indicating that, in the small excess-rate-probability regime, the approximation to the fundamental limit of the compression rate suggested by these bounds is significantly more accurate than the approximations provided by either normal approximation or error exponents. The new bounds reinforce the crucial operational conclusion that, in applications where the blocklength is relatively short and where stringent guarantees are required on the rate, the best achievable rate is no longer close to the entropy. Rather, it is an appropriate, more pragmatic rate, determined via the inverse error exponent function and the blocklength.