Operator-Based Machine Intelligence: A Hilbert Space Framework for Spectral Learning and Symbolic Reasoning

Abstract

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.