In this paper, we study rate-distortion theory for general sources with an emphasis on the existence of optimal reconstruction distributions. Classical existence results rely on compactness assumptions with continuous distortion that are often violated in general settings. By introducing the concentration-compactness principle into the analysis of the rate-distortion functional, we establish the existence of optimal reconstructions under mild coercivity and lower semi-continuity conditions on the distortion function. Our results provide a unified and transparent existence theorem for rate-distortion problems with lower semi-continuous distortion.

翻译：本文研究一般信源的率失真理论，重点探讨最优重建分布的存在性问题。经典存在性结果依赖于失真函数连续性的紧性假设，这在一般设定下常不成立。通过将集中紧性原理引入率失真泛函的分析，我们在失真函数满足温和强制性与下半连续性条件下，建立了最优重建分布的存在性。所得结果为具有下半连续失真的率失真问题提供了一个统一且明晰的存在性定理。