Speaker
Description
Incremental design for computer experiments traditionally relies on space-filling or uniformity criteria, but these become impractical in high dimensions. The greedy minimisation of the $L_s$-mean quantisation error (or distortion) offers a valuable alternative, though it is also computationally intractable for large $d$.
This talk focuses on random designs composed of i.i.d. points sampled from a specific distribution. For large $d$, observing the behaviour predicted by Zador’s theorem requires an impractically large sample size $n$, growing super-exponentially with $d$. We address this challenge by analysing the quantisation problem for spherically symmetric distributions. Our results show that for moderate $n$ random quantisers uniformly distributed on a sphere of suitable radius $R$ achieve exceptional performance. The expected distortion, computed exactly via a triple integral, allows numerical optimisation of $R$. Using extreme-value theory, we also derive approximations for $R$, revealing that, when $n$ grows with $d$ and $d \to \infty$, $R$ may converge to zero or approach a limiting value $R_\infty$ independent of $s$, depending on the growth rate of $n$.
While spherically symmetric distributions may seem restrictive, they provide a starting point for further applications, such as quantising the uniform measure on the hypercube $X=[-1,1]^d$. For large $d$, the uniform measure on $X$ can be approximated by a spherically symmetric distribution, and one can consider random quantisers distributed according to a product measure, which can itself be approximated by a spherically symmetric distribution. Preliminary results suggest that quantisers distributed on the vertices of a smaller hypercube exhibit promising performance.
| Classification | Mainly methodology |
|---|---|
| Keywords | space-filling design, quantisation, distortion, spherically symmetric distribution, Zador's theorem, extreme-value theory |