Exploiting an interplay between norms to analyze scalar quantization schemes

Quantization is intrinsic to several data acquisition systems. This process is especially important in distributed settings, where observations must first be compressed before they are disseminated. There have been many practical successes in the area of quantization, including the acclaimed Lloyd-Max algorithm. This article adopts a different perspective and it explores quantization at a fundamental level, seeking to identify classes of problems for which efficient quantization is possible. The focus is primarily on positive random variables of unbounded support, where severe degradation may occur. Established properties of Banach spaces are exploited, together with the boundedness of probability measures, to prove that efficient quantization schemes necessarily exist in the fine-quantization regime. The results are algorithmic in nature and provide bounds on the number of bits necessary to achieve a desired level of performance.