Artificial neural networks on graded vector spaces

2024-07-26

arXiv.org

AI Summary

This work addresses the challenge that classical neural networks struggle to model hierarchical, structured data arising in algebraic geometry and theoretical physics. We propose the first structure-preserving neural network framework specifically designed for graded vector spaces. Methodologically, we introduce grade-sensitive neurons, equivariant graded layers, and graded-structure-preserving activation functions; further, we develop a representation-theoretic architectural design paradigm and a graded-compatible loss function, integrating graded algebra, group actions, and equivariant learning theory. Our contribution lies in systematically bridging the theoretical gap between algebraic structure and deep learning. Empirical evaluation on invariant prediction over weighted projective spaces and modeling of supersymmetric systems demonstrates substantial improvements over standard neural networks—validating the dual advantages of structural awareness in both generalization performance and physical interpretability.

Technology Category

• Machine Learning: Structured Learning
• Computer Vision: Representation Learning for Vision
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

This paper presents a transformative framework for artificial neural networks over graded vector spaces, tailored to model hierarchical and structured data in fields like algebraic geometry and physics. By exploiting the algebraic properties of graded vector spaces, where features carry distinct weights, we extend classical neural networks with graded neurons, layers, and activation functions that preserve structural integrity. Grounded in group actions, representation theory, and graded algebra, our approach combines theoretical rigor with practical utility. We introduce graded neural architectures, loss functions prioritizing graded components, and equivariant extensions adaptable to diverse gradings. Case studies validate the framework's effectiveness, outperforming standard neural networks in tasks such as predicting invariants in weighted projective spaces and modeling supersymmetric systems. This work establishes a new frontier in machine learning, merging mathematical sophistication with interdisciplinary applications. Future challenges, including computational scalability and finite field extensions, offer rich opportunities for advancing this paradigm.

Problem

• Extends neural networks to graded vector spaces for hierarchical data
• Introduces graded architectures preserving structural integrity in models
• Validates framework in tasks like predicting weighted projective invariants

Innovation

• Graded neural networks preserve structural integrity
• Loss functions prioritize graded components
• Equivariant extensions adapt to diverse gradings