AI 中文总结
该研究分析非参数回归中纯随机划分估计量的逐点收敛性，证明Mondrian树具备形状正则性可达到极小极大速率，得到Proto-NN的逐点集中界并解决相关开放问题，还表明OptiNet的性能更优。
AI 中文摘要
我们研究非参数回归中纯随机划分估计量的逐点收敛速率，其中划分通过纯随机树划分为超矩形，或通过原型规则划分为Voronoi单元，且划分与响应变量独立构建。我们的分析基于单一几何准则——形状正则性，该准则将单元的直径与其体积关联起来，据Bettinger、Portier和Saumard（2026）的研究，在对数因子范围内，形状正则性是达到极小极大速率$n^{-1/(d+2)}$的充要条件。我们证明，中心化均匀树不具备形状正则性——其单元的纵横比随分裂次数呈指数增长，且概率有界远离零——这解释了其误差界中出现的超对数修正项；而Mondrian树的分裂会适配当前单元的几何结构，在概率意义下具备形状正则性，且能达到极小极大速率。将相同分析应用于Voronoi划分，得到了Proto-NN的首个逐点集中界，解决了Györfi和Weiss（2021）提出的开放问题，同时表明OptiNet凭借其η-网构造，以显著更优的成功概率达到极小极大速率，在参数选择合适时甚至几乎必然达到该速率。
英文摘要
We study pointwise convergence rates of purely random partition estimators in nonparametric regression, where the partition -- into hyper-rectangles by purely random trees, or into Voronoi cells by prototype rules -- is built independently of the responses. Our analysis rests on a single geometric criterion, shape regularity, relating the diameter of a cell to its volume, which is shown by Bettinger, Portier and Saumard (2026) to be necessary and sufficient, up to logarithmic factors, for achieving the minimax rate $n^{-1/(d+2)}$. We show that centered and uniform trees are not shape-regular -- their cells' aspect ratio grows exponentially with the number of splits with probability bounded away from zero -- explaining the super-logarithmic corrections in their error bounds, whereas Mondrian trees, whose splits adapt to the current cell geometry, are shape-regular in probability and attain the minimax rate. The same analysis applied to Voronoi partitions yields the first pointwise concentration bounds for Proto-NN, resolving an open problem of Györfi and Weiss (2021), and shows that OptiNet achieves the minimax rate with markedly better success probability -- even almost surely, for a suitable choice of parameters -- thanks to its $η$-net construction.
