Operator-Based Machine Intelligence: A Hilbert Space Framework

This document presents an alternative paradigm for machine learning, shifting from traditional neural networks to a framework rooted in infinite-dimensional Hilbert spaces. It explores how learning tasks can be reinterpreted as operator estimation and computation within these spaces, leveraging tools from functional analysis, signal processing, and spectral theory. The text highlights the advantages of this approach, such as enhanced interpretability, compactness, and theoretical guarantees, while also discussing practical limitations and future research directions to improve scalability and adaptability. Examples like scattering networks and Koopman operator learning are provided to illustrate the empirical competitiveness of this Hilbert space methodology.