Non-asymptotic quantisation of spherically symmetric distributions

TL;DR

This paper analyzes non-asymptotic quantisation of spherically symmetric distributions, showing that moderate sample sizes can achieve high performance and deriving practical methods for optimal radius selection.

Contribution

It provides a detailed analysis of quantisation for spherical distributions, including explicit formulas and numerical methods for optimal radius determination.

Findings

Expected distortion can be computed with arbitrary precision.

Optimal radius r can be efficiently numerically determined.

Depending on n, r converges to zero or a limit independent of s.

Abstract

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 $R^d$ and the decay rate of the associated $L_s$-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 $s$.