Rate–Distortion Limits for Task-Oriented Compression with Side Information

This paper analyzes the semantic rate–distortion problem motivated by task-oriented data compression with side information. The semantic information related to a task is not directly accessible to the encoder but implicitly impacts the observations through a joint probability distribution. The decoder aims to simultaneously recover the observation and infer the semantic information under certain distortion constraints. Notably, this paper advances the related research by involving side information and the observation of two semantic segments at both the encoder and decoder, which significantly complicates the theoretic analysis. We establish the information-theoretic limits for the tradeoff between compression rates and distortions by fully characterizing the rate–distortion function. Additionally, we explicitly derive the corresponding rate–distortion functions under specific Markov conditions for two scenarios: (i) the task is a binary classification of an integer observation as even and odd; and (ii) Gaussian-correlated task and observation. Furthermore, we validate the information-theoretic analysis by conducting a classification-oriented lossy image compression based on deep learning. The results are consistent with theoretical expectations, demonstrating the effectiveness of side information on both distortion and classification accuracy and the rationality of semantic segmentation.