高斯索伯列夫算子的近最优学习
Near-Optimal Learning of Gaussian Sobolev Operators
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中文总结 AI 辅助
研究如何在合理时间内设计有近似保证的替代算子，提出埃尔米特主成分分析近似算法学习高斯索伯列夫算子，该算法计算高效且具谱性质，通过误差分析和数值实验验证其有效性，实现近最优样本复杂性。
中文摘要 AI 辅助
算子学习中的一个关键问题是如何在合理的计算时间内设计出具有可证明近似保证的替代算子。虽然光滑算子可以有效逼近，即随着训练数据量至少有代数收敛，但学习有限正则算子效率较低，原因是样本复杂性的内在诅咒，其样本复杂性速率仅为亚代数。因此，开发能达到这些速率的算法尤为重要。本文提出一种完全数据驱动的算法——埃尔米特主成分分析近似，用于学习具有近最优样本复杂性的高斯索伯列夫算子。它采用主成分分析和加权最小二乘法，计算效率高，且具有谱性质，索伯列夫正则性越高收敛越快。我们对该算法进行了全面误差分析，并通过数值实验验证了理论结果，证实了其在学习索伯列夫算子方面的有效性。
英文摘要
A key question in operator learning is how to design surrogate operators with provable approximation guarantees in reasonable computational time. Whereas smooth operators can be approximated efficiently, i.e., with at least algebraic convergence in the amount of training data, learning finitely regular operators is known to be less efficient. The reason is an intrinsic curse of sample complexity, which allows only subalgebraic sample complexity rates. This fact makes it all the more important to develop algorithms which provably achieve these rates. In this work, we present a fully data-driven algorithm, termed Hermite-PCA approximation, for learning Gaussian Sobolev operators with near-optimal sample complexity. It employs principal component analysis and weighted least-squares methods and is therefore computationally efficient. Moreover, it is spectral, in the sense that it achieves faster (and near-optimal) convergence the higher the Sobolev regularity. We provide a full error analysis of this algorithm, taking into account all sources of error, along with numerical experiments that verify our theoretical results and empirically confirm the efficacy of Hermite-PCA approximation for learning Sobolev operators.
