发表机构
McMaster University; The Vector Institute; The Fields Institute’s Centre for Mathematical AI; École Polytechnique Fédérale de Lausanne (EPFL)（麦克马斯特大学; Vector研究所; 菲尔兹研究所数学人工智能中心; 洛桑联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对$C^{s,1}$函数提出计算高效的闭式延拓公式，满足与样本量无关的界，在$s\boldsymbol{\text{≥}}2$时为新公式，可实现线性存储与对数并行评估时间，改进了相关初始化界。
AI 中文摘要
我们针对$C^{s,1}$函数$f:\boldsymbol{\text{R}}^d\to\boldsymbol{\text{R}}$，在$[0,1]^d$内的$N$个不同点上，识别出用于插值其精确值直至$s$阶导数的显式闭式及变分公式。该重构满足全局$C^{s,1}$半范数和$[0,1]^d$上 Lipschitz 常数的界，且与样本量$N$无关，后者是严格的 Whitney 条件。对于$s\boldsymbol{\text{≥}}2$，该重构在数据点外为实解析函数，且可在 o-minimal 结构$\boldsymbol{\text{R}}_{\text{exp}}$中定义。我们的非线性延拓公式在$s=0$时与 McShane（1934）的公式一致，在$s=1$时与 Le Gruyer 和 Phan（2015）以及 Azagra、Le Gruyer 和 Mudarra（2018）的公式一致，而对于$s\boldsymbol{\text{≥}}2$则是新的。对于固定的$d$和$s\boldsymbol{\text{≥}}2$，我们的闭式公式可通过仅使用基本一元和二元实数运算的电路在数据点外计算，该电路具有$\boldsymbol{\text{O}}(N)$个门和$\boldsymbol{\text{O}}(\text{log}N)$深度，由此产生$\boldsymbol{\text{O}}(N)$的存储量和$\boldsymbol{\text{O}}(\text{log}N)$的并行评估时间。在提供的系数和权重下，该电路可在精确实数 word-RAM 模型中通过$\boldsymbol{\text{O}}(N)$的一次性工作编译。因此，在该提供数据的场景中，我们的非线性构造在初始化界上比 Fefferman 和 Klartag（2009）的$\boldsymbol{\text{O}}(N\text{log}N)$改进了一个对数因子，同时保留了线性存储并通过并行评估达到了对数查询时间。
英文摘要
We identify explicit closed-form and variational formulae for interpolating the exact values and derivatives through order $s$ of a $C^{s,1}$ function $f:\mathbb{R}^d\to\mathbb{R}$ at $N$ distinct points in $[0,1]^d$. The reconstruction satisfies bounds on its global $C^{s,1}$ seminorm and its Lipschitz constant on $[0,1]^d$ that are independent of the sample size $N$, the latter being the sharp Whitney condition. For $s\ge2$, the reconstruction is real analytic away from the data points and definable in the o-minimal structure $\mathbb{R}_{\exp}$. Our nonlinear extension formula coincides with the formula of McShane (1934) for $s=0$ and with the formulae of Le Gruyer and Phan (2015) and Azagra, Le Gruyer, and Mudarra (2018) for $s=1$, and is new for $s\ge2$. For fixed $d$ and $s\ge2$, our closed-form formula is computable away from the data points by a circuit using only elementary unary and binary real operations, with $\mathcal{O}(N)$ gates and $\mathcal{O}(\log N)$ depth. This yields $\mathcal{O}(N)$ storage and $\mathcal{O}(\log N)$ parallel evaluation time. From the supplied coefficients and weights, the circuit can be compiled with $\mathcal{O}(N)$ one-time work in the exact-real word-RAM model. Thus, in this supplied-data setting, our nonlinear construction improves by a logarithmic factor on the $\mathcal{O}(N\log N)$ initialization bound of Fefferman and Klartag (2009), while retaining linear storage and attaining logarithmic query time through parallel evaluation.
Comments10 Pages + Proofs
