Non-asymptotic quantisation of spherically symmetric distributions

Zador's celebrated theorem is a cornerstone of optimal quantisation, establishing both the weak limit of the empirical distribution of an n-point optimal quantiser in Rd and the decay rate of the associated Ls-mean quantisation error. However, for large dimensions d, observing this asymptotic behaviour demands an astronomically large sample size n, which grows super-exponentially with d. Through a detailed analysis of the quantisation problem for spherically symmetric distributions, we demonstrate that for moderate n random quantisers uniformly distributed on a sphere of suitable radius r achieve exceptional performance. The expected distortion, expressed as a triple integral, can be computed with arbitrary precision, and the optimal radius r can be efficiently determined numerically. Leveraging results from extreme-value theory, we derive approximations for r, particularly in scenarios where n scales with d. Depending on the growth rate of n, r may either converge to zero or approach a limiting value that is independent of s.