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Stable Source Coding

Zhenduo Wen, Amin Gohari

TL;DR

This work investigates lossless source coding under an explicit stability constraint, where a small input perturbation (measured by $d_I$) must induce only a bounded change in the codeword (measured by $d_H$). The authors recast stability into a graph-homomorphism problem between a source-graph on a fixed type class and a codeword-graph on the Hamming cube, deriving necessary information-theoretic limits on the rate. In the linear regime, they establish two complementary converse bounds: a maximum-degree bound $F(d_1,p)\le G(d_2,R)$ and a clique-size bound $h_2(d_1/2)\le\Psi(d_2,R)$, together with a sublinear-regime bound that enforces $D_n'\ge D_n$ for constant stability scales. These bounds are demonstrated via reductions to one-type classes, graph constructions, and classical combinatorial results (Ahlswede–Khachatrian, Kleitman), with numerical evaluations illustrating the relative tightness of the bounds. The results illuminate fundamental limits for stable encoders and suggest directions for further refinements using independence numbers and homomorphism-based techniques.

Abstract

A source encoder is stable if a small change in the source sequence (e.g., changing a few symbols) results in a small (or bounded) change in the output codeword. By this definition, the common technique of random binning is unstable; because the mapping is random, two nearly identical source sequences can be assigned to completely unrelated bin indices. We study compression rates of stable lossless source codes. Using combinatorial arguments, we derive information-theoretic limits on the achievable rate as a function of the stability parameters.

Stable Source Coding

TL;DR

This work investigates lossless source coding under an explicit stability constraint, where a small input perturbation (measured by ) must induce only a bounded change in the codeword (measured by ). The authors recast stability into a graph-homomorphism problem between a source-graph on a fixed type class and a codeword-graph on the Hamming cube, deriving necessary information-theoretic limits on the rate. In the linear regime, they establish two complementary converse bounds: a maximum-degree bound and a clique-size bound , together with a sublinear-regime bound that enforces for constant stability scales. These bounds are demonstrated via reductions to one-type classes, graph constructions, and classical combinatorial results (Ahlswede–Khachatrian, Kleitman), with numerical evaluations illustrating the relative tightness of the bounds. The results illuminate fundamental limits for stable encoders and suggest directions for further refinements using independence numbers and homomorphism-based techniques.

Abstract

A source encoder is stable if a small change in the source sequence (e.g., changing a few symbols) results in a small (or bounded) change in the output codeword. By this definition, the common technique of random binning is unstable; because the mapping is random, two nearly identical source sequences can be assigned to completely unrelated bin indices. We study compression rates of stable lossless source codes. Using combinatorial arguments, we derive information-theoretic limits on the achievable rate as a function of the stability parameters.
Paper Structure (15 sections, 7 theorems, 64 equations, 1 figure)

This paper contains 15 sections, 7 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Setup
  4. Notations
  5. Converse bounds
  6. Proofs
  7. Reduction to one type class
  8. Source and codeword graphs and their homomorphism
  9. Relating $\Delta(G'_n)$ and $\omega(G'_n)$ to $\Delta(G_n)$ and $\omega(G_n)$
  10. A maximum degree converse bound
  11. A clique size converse bound
  12. Numerical evaluations
  13. Future work
  14. Proof of Proposition \ref{['prop:reduction-one-type']}
  15. Proof of Lemma \ref{['lem:deg-preserve-simple']}

Key Result

Theorem 1

Define Then, for all $(nd_1,nd_2)$-achievable rates $R$, the following must hold: $F(d_1, p)\ \le\ G(d_2, R).$

Figures (1)

  • Figure :

Theorems & Definitions (12)

  • Theorem 1: Degree bound in the linear regime
  • Theorem 2: Clique size bound in the linear regime
  • Proposition 1: Sublinear regime
  • Proposition 2: Reduction to one type class
  • Definition 1: Source neighborhood graph
  • Definition 2: Codeword neighborhood graph
  • Lemma 1: Maximum degree and maximum clique size of a large induced subgraph
  • proof : Proof of Theorem \ref{['thm::degree_linear']}
  • Theorem 3: Ahlswede & Khachatrian ahlswede1997complete, 1997
  • Theorem 4: Kleitman kleitman1966combinatorial, 1966
  • ...and 2 more