# On the Rate-Distortion Theory for Task-Specific Semantic Communication
###### Abstract  
Semantic communication has attracted considerable interest due to its potential to support emerging human-centric services, such as holographic communications, extended reality (XR), and human-machine interactions. Different from traditional communication systems that focus on minimizing the symbol-level distortion (e.g., bit error rate, signal-to-noise ratio, etc.), semantic communication targets at delivering the intended meaning at the destination user which is often quantified by various statistical divergences, often referred to as the semantic distances. Currently, there still lacks a unified framework to quantify the rate-distortion tradeoff for semantic communication with different task-specific semantic distance measures. To tackle this problem, we propose the task-specific rate-distortion theory for semantic communication where different task-specific statistic divergence metrics can be considered. To investigate the impact of different semantic distance measures on the achievable rate, we consider two popular tasks, classification and signal generation. We present the closed-form expressions of the semantic rate-distortion functions for these two different tasks and compare their performance under various scenarios. Extensive experimental results are presented to verify our theoretical results.  
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inference [10]. Also, in generation tasks, the Wasserstein distance is known to exhibit good performance in capturing the geometric relationships between high-dimensional distributions and providing stable gradient flows for generative adversarial networks (GANs) [11].  
Despite its promising potential, most existing work focuses on optimizing the semantic communication system based on a single predefined and fixed semantic distance metric [12, 13]. There is still lacking a unified information theoretic framework that quantifies the achievable rate for semantic communication under different tasks with different semantic distance measures. In this paper, we propose a novel task-specific rate-distortion theory for semantic communication in which different task-specific statistic divergence-based distance metrics are considered and compared. We consider two popular tasks, classification and signal generation, to investigate the impact of different types of semantic distances on the minimum achievable rate. We present the closed-form expressions of the semantic rate-distortion functions for both tasks and compare their performance under various scenarios. We summarize the main contribution of this paper as follows:  
1. We propose a novel tasks-specific semantic communication architecture that generalizes traditional rate-distortion theory by incorporating a general divergence metrics to quantify semantic distance.
2. We derive the closed-form expressions for the semantic rate-distortion functions under Gaussian semantic sources, specifically for Wasserstein distance, KL divergence, and reverse KL divergence, revealing fundamental tradeoffs among transmission rate, distortion, and semantic distance.
3. Extensive experiments are conducted on image-based semantic communication systems for both generation and classification tasks. Our results suggest that the proposed framework significantly outperforms traditional MSE-based approaches, with reverse KL divergence demonstrating superior perceptual quality in generation tasks and KL divergence achieving higher classification accuracy.  
The rest of this paper is organized as follows: Section 2 introduces the background and preliminaries. The system model and theoretical results are discussed in Sections 3 and 4, respectively. Section 5 present experimental results and the paper is concluded in Section 6.