Practical applications of Set Shaping Theory to Non-Uniform Sequences

TL;DR

This paper demonstrates that Set Shaping Theory can be effectively applied to non-uniform sequences using approximate ordering, achieving theoretical shaping gains and extending previous uniform sequence results.

Contribution

It introduces an approximate ordering method for SST on non-uniform sequences, overcoming exponential complexity and confirming the theory's practical applicability.

Findings

Shaping gain persists for non-uniform sequences.

Approximate ordering preserves SST structural requirements.

Software implementation is publicly available for reproducibility.

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