Optimal random quantisers for spherically symmetric distributions

Key points

• 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.1
• 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.1
• 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 $n\to\infty$ for fixed $d$.1
• For $s=2$, both the optimal radius and the associated minimum expected distortion admit exact expressions.1

Sources (1)

• [1]Optimal random quantisers for spherically symmetric distributions
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 8, 11:53 AM

  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 $L_s$-mean quantisation error.

  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.

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