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

• URL: http://arxiv.org/abs/2507.21189v1
• Date: Sun, 27 Jul 2025 18:52:10 GMT
• Title: Operator-Based Machine Intelligence: A Hilbert Space Framework for Spectral Learning and Symbolic Reasoning
• Authors: Andrew Kiruluta, Andreas Lemos, Priscilla Burity,
• Abstract summary: This report explores an alternative formulation where learning tasks are expressed as sampling and computation in infinite dimensional Hilbert spaces.<n>We present a rigorous mathematical formulation of learning in Hilbert spaces, highlight recent models based on scattering transforms and Koopman operators.<n>The report concludes by outlining directions for scalable and interpretable machine learning grounded in Hilbertian signal processing.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Hilbert Neural Operator: Operator Learning in the Analytic Signal Domain [0.0]
  We introduce the textbfHilbert Neural Operator (HNO), a new neural operator architecture to address some advantages.<n>HNO operates by first mapping the input signal to its analytic representation via the Hilbert transform.<n>We hypothesize that this architecture enables HNO to model operators more effectively for causal, phase-sensitive, and non-stationary systems.
  arXiv  Detail & Related papers  (2025-08-06T21:12:15Z)
• Quantum Spectral Reasoning: A Non-Neural Architecture for Interpretable Machine Learning [0.0]
  We propose a novel machine learning architecture that departs from conventional neural network paradigms.<n>We use quantum spectral methods, specifically Pade approximants and the Lanczos algorithm, for interpretable signal analysis and symbolic reasoning.<n>Our results show that this spectral-symbolic architecture achieves competitive accuracy while maintaining interpretability and data efficiency.
  arXiv  Detail & Related papers  (2025-08-05T07:16:45Z)
• Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning [73.18052192964349]
  We develop a theoretical framework that explains how discrete symbolic structures can emerge naturally from continuous neural network training dynamics.<n>By lifting neural parameters to a measure space and modeling training as Wasserstein gradient flow, we show that under geometric constraints, the parameter measure $mu_t$ undergoes two concurrent phenomena.
  arXiv  Detail & Related papers  (2025-06-26T22:40:30Z)
• A Mathematical Analysis of Neural Operator Behaviors [0.0]
  This paper presents a rigorous framework for analyzing the behaviors of neural operators. We focus on their stability, convergence, clustering dynamics, universality, and generalization error. We aim to offer clear and unified guidance in a single setting for the future design of neural operator-based methods.
  arXiv  Detail & Related papers  (2024-10-28T19:38:53Z)
• Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Space [54.13671100638092]
  Holistic Physics Mixer (HPM) is a framework for integrating spectral and physical information in a unified space.<n>We show that HPM consistently outperforms state-of-the-art methods in both accuracy and computational efficiency.
  arXiv  Detail & Related papers  (2024-10-15T08:19:39Z)
• DimOL: Dimensional Awareness as A New 'Dimension' in Operator Learning [60.58067866537143]
  We introduce DimOL (Dimension-aware Operator Learning), drawing insights from dimensional analysis.<n>To implement DimOL, we propose the ProdLayer, which can be seamlessly integrated into FNO-based and Transformer-based PDE solvers.<n> Empirically, DimOL models achieve up to 48% performance gain within the PDE datasets.
  arXiv  Detail & Related papers  (2024-10-08T10:48:50Z)
• Mechanistic Neural Networks for Scientific Machine Learning [58.99592521721158]
  We present Mechanistic Neural Networks, a neural network design for machine learning applications in the sciences. It incorporates a new Mechanistic Block in standard architectures to explicitly learn governing differential equations as representations. Central to our approach is a novel Relaxed Linear Programming solver (NeuRLP) inspired by a technique that reduces solving linear ODEs to solving linear programs.
  arXiv  Detail & Related papers  (2024-02-20T15:23:24Z)
• A Spectral Approach for Learning Spatiotemporal Neural Differential Equations [0.0]
  We propose a neural-ODE based method that uses spectral expansions in space to learn unbounded differential equations. By developing a spectral framework for learning both PDEs and integro-differential equations we extend machine learning methods to apply to DEs and a larger class of problems.
  arXiv  Detail & Related papers  (2023-09-28T03:22:49Z)
• Minimax Optimal Kernel Operator Learning via Multilevel Training [11.36492861074981]
  We study the statistical limit of learning a Hilbert-Schmidt operator between two infinite-dimensional Sobolev reproducing kernel Hilbert spaces. We develop a multilevel kernel operator learning algorithm that is optimal when learning linear operators between infinite-dimensional function spaces.
  arXiv  Detail & Related papers  (2022-09-28T21:31:43Z)
• Reinforcement Learning from Partial Observation: Linear Function Approximation with Provable Sample Efficiency [111.83670279016599]
  We study reinforcement learning for partially observed decision processes (POMDPs) with infinite observation and state spaces. We make the first attempt at partial observability and function approximation for a class of POMDPs with a linear structure.
  arXiv  Detail & Related papers  (2022-04-20T21:15:38Z)
• A Free Lunch from the Noise: Provable and Practical Exploration for Representation Learning [55.048010996144036]
  We show that under some noise assumption, we can obtain the linear spectral feature of its corresponding Markov transition operator in closed-form for free. We propose Spectral Dynamics Embedding (SPEDE), which breaks the trade-off and completes optimistic exploration for representation learning by exploiting the structure of the noise.
  arXiv  Detail & Related papers  (2021-11-22T19:24:57Z)
• Non-parametric Active Learning and Rate Reduction in Many-body Hilbert Space with Rescaled Logarithmic Fidelity [4.781805457699204]
  In quantum and quantum-inspired machine learning, the very first step is to embed the data in quantum space known as Hilbert space. We propose the rescaled logarithmic fidelity (RLF) and a non-parametric active learning in the quantum space, which we name as RLF-NAL. Our results imply that the machine learning in the Hilbert space complies with the principles of maximal coding rate reduction.
  arXiv  Detail & Related papers  (2021-07-01T03:13:16Z)
• Transforming Feature Space to Interpret Machine Learning Models [91.62936410696409]
  This contribution proposes a novel approach that interprets machine-learning models through the lens of feature space transformations. It can be used to enhance unconditional as well as conditional post-hoc diagnostic tools. A case study on remote-sensing landcover classification with 46 features is used to demonstrate the potential of the proposed approach.
  arXiv  Detail & Related papers  (2021-04-09T10:48:11Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.