证明了任意连续泛函可用线性测量加标量非线性实现统一逼近。
Universal Approximation of Continuous Functionals on Compact Subsets via Linear Measurements and Scalar Nonlinearities
- 先做有限线性测量，再用标量非线性组合
- 可在紧集上对任意连续泛函实现一致逼近
- 为算子学习和成像设计提供理论支持
我们研究了希尔伯特空间乘积中紧子集上连续泛函的统一逼近问题。证明了任意此类泛函均可通过有限个连续线性测量与连续标量非线性组合的模型进行一致逼近。同时将该逼近原理扩展至巴拿赫空间取值映射，得到有限秩逼近结果。这些结论为算子学习与成像中常见的“测量、应用标量非线性、再组合”设计模式提供了紧集上的理论依据。
原文摘要 · Abstract (English)
We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear measurements of the inputs and then combine these measurements through continuous scalar nonlinearities. We also extend the approximation principle to maps with values in a Banach space, yielding finite-rank approximations. These results provide a compact-set justification for the common ``measure, apply scalar nonlinearities, then combine'' design pattern used in operator learning and imaging.
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