Rate–Distortion Theory for Task-Oriented Distributed Cross-Modal Source Coding

Multi-modal traffic is becoming dominant across vertical domains. Rather than reconstructing raw data with content-agnostic fidelity, many emerging applications are inherentlytask-orientedand exhibit exploitable inter-modal correlations. In this paper we study such scenarios within the Shannon framework, modeling them as a CEO-type multi-terminal source coding problem under logarithmic loss, where asemantic priorSis available at both the encoders and the decoder. We refer to this formulation astask-oriented distributed cross-modal source coding(TD-CMSC) to emphasize its multimodal sensing application, while the underlying mathematical problem remains a classical log-loss CEO / multi-terminal source coding model augmented with the semantic prior S. On the theoretical side, we characterize the corresponding rate–distortion region under log-loss with semantic prior S in a general multi-encoder setting, and, under a total-rate constraint, we derive the associated distortion– rate function together with an explicit single-letter expression for the optimal per-modality rate allocation at the extreme points of the region. On the technical side, leveraging these results, we design a coding system for vector Gaussian sources that combines task-oriented conditional quantization with successiveWyner–Ziv coding. Numerical experiments confirm the feasibility and rate efficiency of the proposed design.