Problem
Research questions and friction points this paper is trying to address.
Whitney extension
interpolation
C^{s,1} function
closed-form formula
computational efficiency
A Computationally-Efficient Closed-Form $C^{s,1}$-Extension Formula
🤖 AI Summary
This study addresses the high computational complexity of high-dimensional interpolation by efficiently reconstructing $C^{s,1}$ functions satisfying Whitney conditions. Methodologically, it introduces the first explicit closed-form formulas for $s \ge 2$, unifying and extending classical theory while overcoming the bottleneck of linear initialization. This enables analytical reconstruction beyond data points with a globally bounded seminorm independent of sample size. Furthermore, the computational pipeline is optimized by integrating real arithmetic circuits, parallel evaluation, and exact Real RAM compilation techniques. Ultimately, the proposed approach achieves $O(N)$ storage and $O(\log N)$ parallel query time, eliminating the logarithmic factor present in existing algorithms and significantly enhancing overall efficiency.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Closed-form extension formula
$C^{s,1}$ interpolation
Computational efficiency
Whitney condition
Parallel evaluation
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