# Graded Neural Networks

arXiv: https://arxiv.org/abs/2502.17751 · HTML: https://mlsanity.com/paper/2502.17751/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 17 references that are in the map.
- First posted on arXiv 25 Feb 2025.

## Builds on (most cited first, 4 of 4)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Machine learning for moduli space of genus two curves and an application to isogeny based cryptography](https://mlsanity.com/paper/2403.17250/) (2024)
- [Artificial Neural Networks on Graded Vector Spaces](https://mlsanity.com/paper/2407.19031/) (2024)

## Built on by (most cited first, 4 of 4)

- [Internalizing Tools as Morphisms in Graded Transformers](https://mlsanity.com/paper/2511.17840/) (2025)
- [Graded Transformers](https://mlsanity.com/paper/2507.20108/) (2025)
- [From Polynomials to Databases: Arithmetic Structures in Galois Theory](https://mlsanity.com/paper/2511.16622/) (2025)
- [Hierarchical Grading in Large Language Models](https://mlsanity.com/paper/2607.22757/) (2026)

## Abstract (arXiv)

This paper presents a novel framework for graded neural networks (GNNs) built over graded vector spaces V_wn, extending classical neural architectures by incorporating algebraic grading. Leveraging a coordinate-wise grading structure with scalar action λ star x = (λqi xi), defined by a tuple w = (q0, ldots, qn-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.