Rate-distortion Theory on Non-compact Spaces: A Concentration-compactness Approach

Abstract: 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 that are often violated in non-compact 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 conditions on the distortion function. Our results provide a unified and transparent existence theorem for rate-distortion problems on general non-compact spaces.