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

arXiv: https://arxiv.org/abs/2507.21189 · HTML: https://mlsanity.com/paper/2507.21189/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 24 references that are in the map.
- First posted on arXiv 27 Jul 2025.

## Builds on (most cited first, 8 of 8)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Deep Scattering Spectrum](https://mlsanity.com/paper/1304.6763/) (2013)
- [From Pixels and Words to Waves: A Unified Framework for Spectral Dictionary vLLMs](https://mlsanity.com/paper/2506.18943/) (2025)

## Built on by (most cited first, 1 of 1)

- [Schrodinger AI: A Unified Spectral-Dynamical Framework for Classification, Reasoning, and Operator-Based Generalization](https://mlsanity.com/paper/2512.22774/) (2025)

## Abstract (arXiv)

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.