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Single-Letter Characterization of the Mismatched Distortion-Rate Function

Maël Le Treust, Tristan Tomala

TL;DR

The paper resolves the open problem of a single-letter characterization for the mismatched distortion-rate function $C_d^{\star}(R)$. It introduces a time-sharing construction to convexify Lapidoth's bound and shows that only two regime types (high vs. low rate) are needed. The core novelty is a converse built around auxiliary variables $W_t=(M,U^{t-1})$ with rates $R_t=I(U_t;W_t)$, plus Carathéodory-based cardinality bounds yielding $|\mathcal{W}|=|\mathcal{U}|+3$ and $|\mathcal{T}|\le 2$. The results present a complete single-letter description: a convexified objective $C_d^{\star}(R)$ characterized by a set of feasible $(W,V)$ distributions under Markov and information constraints, and the encoder’s optimality is preserved under cardinality reduction. This advances understanding of decentralized coding with mismatched objectives and provides precise tools for evaluating the achievable mismatched distortion-rate trade-off.

Abstract

The mismatched distortion-rate problem has remained open since its formulation by Lapidoth in 1997. In this paper, we characterize the mismatched distortion-rate function. Our single-letter solution highlights the adequate conditional distributions for the encoder and the decoder. The achievability result relies on a time-sharing argument that allows to convexify the upper bound of Lapidoth. We show that it is sufficient to consider two regimes, one with a large rate and another one with a small rate. Our main contribution is the converse proof. Suppose that the encoder selects a single-letter conditional distribution distinct from the one in the solution, we construct an encoding strategy that leads to the same expected cost for both encoder and decoder. This ensures that the encoder cannot gain by changing the single-letter conditional distribution. This argument relies on a careful identification of the sequence of auxiliary random variables. By building on Caratheodory's Theorem we show that the cardinality of the auxiliary random variables is equal to the one of the source alphabet plus three.

Single-Letter Characterization of the Mismatched Distortion-Rate Function

TL;DR

The paper resolves the open problem of a single-letter characterization for the mismatched distortion-rate function . It introduces a time-sharing construction to convexify Lapidoth's bound and shows that only two regime types (high vs. low rate) are needed. The core novelty is a converse built around auxiliary variables with rates , plus Carathéodory-based cardinality bounds yielding and . The results present a complete single-letter description: a convexified objective characterized by a set of feasible distributions under Markov and information constraints, and the encoder’s optimality is preserved under cardinality reduction. This advances understanding of decentralized coding with mismatched objectives and provides precise tools for evaluating the achievable mismatched distortion-rate trade-off.

Abstract

The mismatched distortion-rate problem has remained open since its formulation by Lapidoth in 1997. In this paper, we characterize the mismatched distortion-rate function. Our single-letter solution highlights the adequate conditional distributions for the encoder and the decoder. The achievability result relies on a time-sharing argument that allows to convexify the upper bound of Lapidoth. We show that it is sufficient to consider two regimes, one with a large rate and another one with a small rate. Our main contribution is the converse proof. Suppose that the encoder selects a single-letter conditional distribution distinct from the one in the solution, we construct an encoding strategy that leads to the same expected cost for both encoder and decoder. This ensures that the encoder cannot gain by changing the single-letter conditional distribution. This argument relies on a careful identification of the sequence of auxiliary random variables. By building on Caratheodory's Theorem we show that the cardinality of the auxiliary random variables is equal to the one of the source alphabet plus three.
Paper Structure (11 sections, 6 theorems, 44 equations, 1 figure)

This paper contains 11 sections, 6 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. System Model
  3. Converse Proof of Theorem \ref{['theo:main']}
  4. Identification of the auxiliary random variables
  5. Distribution $\mathcal{Q}^{\sigma,\tau}_{UWV}$ belongs to the set $\mathbb{Q}$
  6. Proof of Lemma \ref{['lemma:UnprofitableDeviation']}
  7. Cardinality bounds for the auxiliary random variables
  8. Proof of Lemma \ref{['lemma:CardW']}
  9. Convex Optimization Problem
  10. Caratheodory reduction
  11. Tie-breaking rule

Key Result

Theorem 1

The mismatched distortion-rate function writes It is the convexification, i.e. the greatest convex minorant, of where $W\in\mathcal{W}$ is an auxiliary random variable with $|\mathcal{W}|= |\mathcal{U}|+3$. The set of compatible distributions$\mathbb{P}(\mathcal{Q}_{WV},\mathsf{R})$, and the set of encoder's deviations$\mathbb{D}(\mathcal{Q}_{W},\mathsf{R})$ are defined by

Figures (1)

  • Figure 1: The mismatched distortion-rate problem.

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Lemma 1
  • proof : Lemma \ref{['lemma:MarkovMarginal']}
  • Lemma 2
  • Lemma 3
  • proof : Lemma \ref{['lemma:rate']}
  • Lemma 4
  • Lemma 5