Non-asymptotic quantisation of spherically symmetric distributions

• URL: http://arxiv.org/abs/2605.12568v1
• Date: Tue, 12 May 2026 10:01:41 GMT
• Title: Non-asymptotic quantisation of spherically symmetric distributions
• Abstract summary: Zador's celebrated theorem is a cornerstone of optimal quantisation.<n>We show that for moderate $n$ random quantisers uniformly distributed on a sphere of suitable radius $r$ achieve exceptional performance.<n>We derive approximations for $r$, particularly in scenarios where $n$ scales with $d$.
• Score: 1.2031796234206138
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $s$.

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