用希尔伯特空间重构机器学习，实现可解释的谱学习与符号推理。
Operator-Based Machine Intelligence: A Hilbert Space Framework for Spectral Learning and Symbolic Reasoning
- 将学习任务建模为无限维希尔伯特空间中的采样与计算
- 结合谱算子、散射变换等工具，提升模型可解释性
- 适合追求理论深度与可解释性的研究人员
传统机器学习模型，尤其是神经网络，基于有限维参数空间和非线性函数逼近。本文探索一种替代框架：将学习任务表达为在无限维希尔伯特空间中的采样与计算，利用泛函分析、信号处理和谱理论工具。回顾了再生核希尔伯特空间（RKHS）、谱算子学习及小波域表示等基础概念。提出了希尔伯特空间中学习的严格数学形式，重点介绍了基于散射变换和科普曼算子的最新模型，并讨论其相对于传统神经架构的优势与局限。最后展望了基于希尔伯特信号处理的可扩展、可解释机器学习发展方向。
原文摘要 · Abstract (English)
Traditional machine learning models, particularly neural networks, are rooted in finite-dimensional parameter spaces and nonlinear function approximations. This report explores an alternative formulation where learning tasks are expressed as sampling and computation in infinite dimensional Hilbert spaces, leveraging tools from functional analysis, signal processing, and spectral theory. We review foundational concepts such as Reproducing Kernel Hilbert Spaces (RKHS), spectral operator learning, and wavelet-domain representations. We present a rigorous mathematical formulation of learning in Hilbert spaces, highlight recent models based on scattering transforms and Koopman operators, and discuss advantages and limitations relative to conventional neural architectures. The report concludes by outlining directions for scalable and interpretable machine learning grounded in Hilbertian signal processing.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。