Graded Neural Networks

TL;DR

This paper introduces graded neural networks (GNNs) that incorporate algebraic grading into neural architectures, providing a new theoretical framework with practical applications in machine learning and photonics.

Contribution

It develops a novel graded neural network framework with algebraic grading, extending classical neural networks and addressing computational challenges.

Findings

Established theoretical properties of graded spaces.

Designed a comprehensive GNN architecture.

Discussed potential applications in machine learning and photonics.

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 $\lambda \star \x = (\lambda^{q_i} x_i)$, 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.