Scalar versus vector quantization: worst case analysis

We study the potential merits of vector quantization and show that there can be an arbitrary discrepancy between the worst case rates required for scalar and vector quantization. Specifically, we describe a random variable and a distortion measure where quantization of a single instance to within a given distortion requires an arbitrarily large number of bits in the worst case, but quantization of multiple independent instances to within the same distortion requires an arbitrarily small number of bits per instance in the worst case. We relate this discrepancy to expander graphs, representation- and cover-numbers of set systems, and a problem studied by Slepian, Wolf, and Wyner (1973).