Asymptotic optimality of single-sphere quantisers among all random quantisers

Determine whether, for spherically concentrated target distributions such as the uniform distribution on the unit ball or a spherically symmetric normal distribution, random quantisers uniformly distributed on a sphere of radius r√(1−1/λ²) are asymptotically optimal among all random quantisers when n^{1/d}→λ with λ>1.

Background

The paper proves asymptotic limits for the optimal radius and distortion only within the family of random quantisers whose points are independently and uniformly distributed on a single sphere. For target measures with norm concentration, the limiting optimal radius in the regime n{1/d}→λ>1 is r√(1−1/λ²), where r is the limiting target radius.

Numerical results for the uniform measure on the unit ball and for a spherically symmetric normal distribution suggest that these single-sphere quantisers may remain optimal even when compared with arbitrary random-quantiser distributions. The paper does not establish this broader optimality claim; proving or disproving it would extend the single-sphere asymptotic theorem to the full class of random quantisers.

References

The numerical evolution of n_1(d,s) in Figure~\ref{F:asymptotics-ball_s2} suggests that, in these two situations and for sufficiently small $>1$, quantisers with distribution $P_{\sqrt{1-1/2}$ may be asymptotically optimal among all random quantisers when $n{1/d}\to$. This conclusion is empirical and is not established by Theorem~\ref{Th:joint-exact-distortion-limit}, which optimises only over single-sphere distributions.

— Optimal random quantisers for spherically symmetric distributions  (2610.11772 - Pronzato et al., 8 Oct 2026) in Section 4.2, immediately following Theorem 4.2