An Indirect Rate-Distortion Characterization for Semantic Sources: General Model and the Case of Gaussian Observation

TL;DR

This paper introduces a new semantic source model with intrinsic and extrinsic parts, analyzes its rate-distortion tradeoff using indirect theory, and provides solutions for Gaussian cases with linear relationships.

Contribution

It proposes a novel semantic source model and derives its rate-distortion function, including explicit solutions for Gaussian observations with linear relations.

Findings

The semantic rate-distortion function is characterized by a convex optimization problem.

A reverse water-filling solution is derived under diagonalizability conditions.

The model captures the tradeoff between intrinsic semantic features and extrinsic observations.

Abstract

A new source model, which consists of an intrinsic state part and an extrinsic observation part, is proposed and its information-theoretic characterization, namely its rate-distortion function, is defined and analyzed. Such a source model is motivated by the recent surge of interest in the semantic aspect of information: the intrinsic state corresponds to the semantic feature of the source, which in general is not observable but can only be inferred from the extrinsic observation. There are two distortion measures, one between the intrinsic state and its reproduction, and the other between the extrinsic observation and its reproduction. Under a given code rate, the tradeoff between these two distortion measures is characterized by the rate-distortion function, which is solved via the indirect rate-distortion theory and is termed as the semantic rate-distortion function of the source. As an application of the general model and its analysis, the case of Gaussian extrinsic observation is studied, assuming a linear relationship between the intrinsic state and the extrinsic observation, under a quadratic distortion structure. The semantic rate-distortion function is shown to be the solution of a convex programming programming problem with respect to an error covariance matrix, and a reverse water-filling type of solution is provided when the model further satisfies a diagonalizability condition.