Single-Letter Characterization of the Mismatched Distortion-Rate Function

TL;DR

This paper provides a single-letter characterization of the mismatched distortion-rate function, resolving an open problem by identifying optimal conditional distributions and proving the converse through a novel encoding strategy.

Contribution

It offers the first single-letter solution to the mismatched distortion-rate problem, including a new converse proof and auxiliary variable cardinality bounds.

Findings

Characterizes the mismatched distortion-rate function.

Identifies sufficient regimes for encoding rates.

Establishes the optimality of specific conditional distributions.

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.