Towards Weak Source Coding

In this paper, the authors provide a weak decoding version of the traditional source coding theorem of Claude Shannon. The central bound that is obtained is $$\chi>\log_{\epsilon}(2^{-n(H(X)+\epsilon)})$$ where $$\chi=\frac{\log(k)}{n(H(X)+\epsilon)}$$ and $k$ is the number of unsupervised learning classes formed out of the non-typical source sequences. The bound leads to the conclusion that if the number of classes is high enough, the reliability function might possibly be improved. The specific regime in which this improvement might be allowable is the one in which the atypical-sequence clusters are small in size and sparsely placed; similar regimes might also show an improvement.