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Graded Neural Networks

Tony Shaska

TL;DR

Graded Neural Networks (GNNs) introduce a principled framework for hierarchical data by endowing neural computations with algebraic grading via $\lambda \star \mathbf{x}=(\lambda^{q_i} x_i)$ on $\mathcal{V}_\mathbf{q}^n$. The approach defines graded neurons, graded ReLU, exponential activations, and graded loss families that respect the underlying grade structure, and it recovers classical neural networks when $\mathbf{q}=(1,\ldots,1)$. The authors establish foundational properties, including convexity of graded losses, universal approximation for graded-homogeneous functions, exact monomial representations, Lipschitz activation behavior, and grade-aware convergence of optimization; they also address numerical stability and computational efficiency. They further demonstrate the framework’s relevance to domains such as algebraic geometry (genus-two invariants), quantum physics (supersymmetric systems), and neuromorphic/photonic hardware, arguing that grading provides a robust, scalable path toward structured deep learning with broad practical impact.

Abstract

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.

Graded Neural Networks

TL;DR

Graded Neural Networks (GNNs) introduce a principled framework for hierarchical data by endowing neural computations with algebraic grading via on . The approach defines graded neurons, graded ReLU, exponential activations, and graded loss families that respect the underlying grade structure, and it recovers classical neural networks when . The authors establish foundational properties, including convexity of graded losses, universal approximation for graded-homogeneous functions, exact monomial representations, Lipschitz activation behavior, and grade-aware convergence of optimization; they also address numerical stability and computational efficiency. They further demonstrate the framework’s relevance to domains such as algebraic geometry (genus-two invariants), quantum physics (supersymmetric systems), and neuromorphic/photonic hardware, arguing that grading provides a robust, scalable path toward structured deep learning with broad practical impact.

Abstract

This paper presents a novel framework for graded neural networks (GNNs) built over graded vector spaces , extending classical neural architectures by incorporating algebraic grading. Leveraging a coordinate-wise grading structure with scalar action , defined by a tuple , 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.

Paper Structure

This paper contains 18 sections, 16 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Graded Vector Spaces
  3. Generalized Gradation
  4. Graded Linear Maps
  5. Operations over Graded Vector Spaces
  6. Inner Graded Vector Spaces and their Norms
  7. Graded and Filtered Structures
  8. Learning from Coarse to Fine.
  9. Graded Neural Networks (GNN)
  10. ReLU Activation
  11. Graded Loss Functions
  12. Optimizers
  13. Theoretical Properties of Graded Neural Networks
  14. Applications and Theoretical Guarantees of Graded Neural Networks
  15. Implementation Challenges
  16. ...and 3 more sections

Key Result

Proposition 1

Let $\mathcal{V}^n_\mathbf{q}(k)$ and $\mathcal{V}^m_{\mathbf{q}'}(k)$ be graded vector spaces with grading vectors $\mathbf{q} = (q_0, \ldots, q_{n-1})$ and $\mathbf{q}' = (r_0, \ldots, r_{m-1})$, respectively, where $q_i, r_j \in \mathbb{Q}_{>0}$. Let $L: \mathcal{V}^n_\mathbf{q}(k) \to \mathcal{V In particular, $L$ is grade-preserving (i.e., $d = 0$) if and only if $a_{ij} \neq 0$ implies $r_i

Theorems & Definitions (49)

  • Example 1
  • Example 2: Moduli Space of Genus 2 Curves
  • Proposition 1
  • proof
  • Example 3
  • Proposition 2
  • proof
  • Corollary 1
  • proof
  • Example 4
  • ...and 39 more