Graded Neural Networks
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917430622683136 |
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| author | Shaska, Tony |
| author_facet | Shaska, Tony |
| contents | This paper presents a novel framework for graded neural networks (GNNs) built over graded vector spaces $\V_\w^n$, extending classical neural architectures by incorporating algebraic grading. Leveraging a coordinate-wise grading structure with scalar action $λ\star \x = (λ^{q_i} x_i)$, defined by a tuple $\w = (q_0, \ldots, q_{n-1})$, we introduce graded neurons, layers, activation functions, and loss functions that adapt to feature significance. Theoretical properties of graded spaces are established, followed by a comprehensive GNN design, addressing computational challenges like numerical stability and gradient scaling. Potential applications span machine learning and photonic systems, exemplified by high-speed laser-based implementations. This work offers a foundational step toward graded computation, unifying mathematical rigor with practical potential, with avenues for future empirical and hardware exploration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17751 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Graded Neural Networks Shaska, Tony Machine Learning Artificial Intelligence 16W50, 13A02, I.2; I.2.6 This paper presents a novel framework for graded neural networks (GNNs) built over graded vector spaces $\V_\w^n$, extending classical neural architectures by incorporating algebraic grading. Leveraging a coordinate-wise grading structure with scalar action $λ\star \x = (λ^{q_i} x_i)$, defined by a tuple $\w = (q_0, \ldots, q_{n-1})$, we introduce graded neurons, layers, activation functions, and loss functions that adapt to feature significance. Theoretical properties of graded spaces are established, followed by a comprehensive GNN design, addressing computational challenges like numerical stability and gradient scaling. Potential applications span machine learning and photonic systems, exemplified by high-speed laser-based implementations. This work offers a foundational step toward graded computation, unifying mathematical rigor with practical potential, with avenues for future empirical and hardware exploration. |
| title | Graded Neural Networks |
| topic | Machine Learning Artificial Intelligence 16W50, 13A02, I.2; I.2.6 |
| url | https://arxiv.org/abs/2502.17751 |