AI 中文总结
针对高维超插值受维度诅咒问题，本文建立内在低塔克秩理论，构建统一理论框架，验证贪婪自适应索引选择方案特性，提出实用算法和管道，通过实验证实超插值系数低秩可压缩性，为其奠定统一结构和数学基础。
AI 中文摘要
高维超插值因维度诅咒而受阻，系数张量随维度指数增长。现有研究多为启发式算法优化，忽视张量固有结构特性。本文建立超插值系数张量的内在低ε-塔克秩严格理论，给出近最优低秩逼近且误差界几乎与维度无关。还构建统一的塔克兼容理论框架，整合两种张量CUR分解，得出仅依赖张量谱特性和索引集几何的紧密稳定的弗罗贝尼乌斯范数误差估计。通过数学验证贪婪自适应索引选择方案的收敛性和数值稳定性并证明其近最优性，实现免构建完整系数阵列的全张量自由超插值工作流程。提出三种实用贪婪TCUR算法和轻量级TCUR到塔克再压缩管道。数值实验验证理论预测，证实超插值系数的内在低秩可压缩性。与以往以算法为中心的研究不同，本文优先进行严格理论刻画而非实现技巧，为高维超插值建立统一结构和数学基础。
英文摘要
High-dimensional hyperinterpolation is severely hampered by the curse of dimensionality, as its coefficient tensors grow exponentially with the ambient dimension. Existing research predominantly focuses on heuristic algorithmic optimizations, often overlooking the inherent structural properties of these tensors. This paper establishes a rigorous theory of intrinsic low-$ε$-Tucker-rank for hyperinterpolation coefficient tensors, delivering near-optimal low-rank approximations with error bounds that are nearly independent of the dimension. We further construct a unified, Tucker-compatible theoretical framework that integrates both Chidori-type and Fiber-type tensor CUR (TCUR) decompositions, deriving tight and stable Frobenius-norm error estimates that depend exclusively on tensor spectral properties and index set geometry. We mathematically verify the convergence and numerical stability of greedy adaptive index selection schemes and prove their near-optimality, enabling a fully tensor-free hyperinterpolation workflow that avoids constructing the full coefficient array. Three practical greedy TCUR algorithms and a lightweight TCUR-to-Tucker recompression pipeline are proposed as direct corollaries of our structural theory. Numerical experiments across three distinct families of high-dimensional test functions validate all theoretical predictions and confirm the intrinsic low-rank compressibility of hyperinterpolation coefficients. In contrast to prior algorithm-centric studies, this work prioritizes rigorous theoretical characterization over implementation tricks, establishing a unified structural and mathematical foundation for high-dimensional hyperinterpolation.
Comments48 pages, 7 figures
