Practical applications of Set Shaping Theory to Non-Uniform Sequences

TL;DR

Set Shaping Theory (SST) investigates bijections $f$ that map a length-$N$ sequence set $X_N$ to a length-$N+K$ set $Y_{N+K}$ with $K>0$, so that the average entropy-length product decreases from $N H_0(s)$ to $(N+K)H_0(y)$. The main contribution is an approximate entropic ordering algorithm with $O(N)$ complexity that enables SST to be applied to non-uniform sources, bypassing the exponential cost of exact ordering. Empirical results show the inequality $(N+K)H_0(y) < N H_0(x)$ holds on average and the shaping gain increases with alphabet size, extending SST beyond uniformly distributed data. The work also ensures reproducibility by releasing publicly available code on GitHub.

Abstract

Set Shaping Theory (SST) moves beyond the classical fixed-space model by constructing bijective mappings the original sequence set into structured regions of a larger sequence space. These shaped subsets are characterized by a reduced average information content, measured by the product of the empirical entropy and the length, yielding (N +k)H0(f(s)) < NH0(s), which represents the universal coding limit when the source distribution is unknown. The principal experimental difficulty in applying Set Shaping Theory to non-uniform sequences arises from the need to order the sequences of both the original and transformed sets according to their information content. An exact ordering of these sets entails exponential complexity, rendering a direct implementation impractical. In this article, we show that this obstacle can be overcome by performing an approximate but informative ordering that preserves the structural requirements of SST while achieving the shaping gain predicted by the theory. This result extends previous experimental findings obtained for uniformly distributed sequences and demonstrates that the shaping advantage of SST persists for non-uniform sequences. Finally, to ensure full reproducibility, the software implementing the proposed method has been made publicly available on GitHub, enabling independent verification of the results reported in this work