Optimal random quantisers for spherically symmetric distributions
Abstract
Zador's celebrated theorem is a cornerstone of optimal quantisation: it establishes both the weak limit of the empirical distribution of an optimal n-point quantiser in R^d and the decay rate of the associated Ls-mean quantisation error. In large dimension, however, observing this asymptotic behaviour requires an astronomically large sample size. We prove that, for spherically symmetric target distributions, optimisation over all spherically symmetric distributions is a convex problem and derive an equivalence theorem that both characterises global optimality and yields a constructive algorithm. We show that, for moderate n, random quantisers uniformly distributed on a sphere of suitably chosen radius R perform exceptionally well and, over a broad range of values of n, are numerically certified to be optimal among all random quantisers. Their expected distortion has an explicit integral representation that can be evaluated to arbitrary precision, and we prove concentration across random quantisers: the distortion variance tends to zero as n → ∞ for fixed d. For s = 2, both the optimal radius and the associated minimum expected distortion admit exact expressions. For general s, the optimal radius can be determined efficiently, and extreme-value theory provides useful approximations when n grows with d. Depending on this growth rate, R either converges to zero or approaches a positive limit that is independent of s.
Related
- Shrinkage Estimation for Spherically Symmetric Distributions under Modified Balanced Loss Functions
- Joint distributions of eigenvectors of symmetric random tensors
- A $π/ 4$ lower bound for random access in symmetric linear ensembles under optimal decoding
- Spherically symmetric Finsler metrics satisfying the σT-condition
- Unbounded log-concavity breaks in independence polynomials of spherically symmetric trees
BibTeX
@misc{pronzato2026optimal,
title = {{Optimal random quantisers for spherically symmetric distributions}},
author = {Luc Pronzato and Anatoly A. Zhigljavsky},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11772},
eprint = {2610.11772},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05786349v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal random quantisers for spherically symmetric distributions
AU - Luc Pronzato
AU - Anatoly A. Zhigljavsky
PY - 2026
DA - 2026/10/08
PB - HAL
DO - 10.48550/arxiv.2610.11772
UR - https://hal.science/hal-05786349v1
AB - Zador's celebrated theorem is a cornerstone of optimal quantisation: it establishes both the weak limit of the empirical distribution of an optimal n-point quantiser in R^d and the decay rate of the associated Ls-mean quantisation error. In large dimension, however, observing this asymptotic behaviour requires an astronomically large sample size. We prove that, for spherically symmetric target distributions, optimisation over all spherically symmetric distributions is a convex problem and derive an equivalence theorem that both characterises global optimality and yields a constructive algorithm. We show that, for moderate n, random quantisers uniformly distributed on a sphere of suitably chosen radius R perform exceptionally well and, over a broad range of values of n, are numerically certified to be optimal among all random quantisers. Their expected distortion has an explicit integral representation that can be evaluated to arbitrary precision, and we prove concentration across random quantisers: the distortion variance tends to zero as n → ∞ for fixed d. For s = 2, both the optimal radius and the associated minimum expected distortion admit exact expressions. For general s, the optimal radius can be determined efficiently, and extreme-value theory provides useful approximations when n grows with d. Depending on this growth rate, R either converges to zero or approaches a positive limit that is independent of s.
ER -