We introduce the Graded Transformer framework, a novel class of sequence models that embeds algebraic inductive biases through grading transformations on vector spaces. Extending the theory of Graded Neural Networks (GNNs), we propose two architectures: the Linearly Graded Transformer (LGT) and the Exponentially Graded Transformer (EGT). These models apply parameterized scaling operators-governed by fixed or learnable grading tuples and, for EGT, exponential factors to infuse hierarchical structure into attention and representation layers, enhancing efficiency for structured data. We derive rigorous theoretical guarantees, including universal approximation theorems for continuous and Sobolev functions, reduced sample complexity via effective VC dimension bounds, Lipschitz continuity of graded operations, and robustness to adversarial perturbations. A graded loss function ensures gradient stability and alignment with domain priors during optimization. By treating grades as differentiable parameters, the framework enables adaptive feature prioritization, overcoming limitations of fixed grades in prior work. The Graded Transformer holds transformative potential for hierarchical learning and neurosymbolic reasoning, with applications spanning algebraic geometry (e.g., moduli spaces and zeta functions), physics (e.g., multiscale simulations), natural language processing (e.g., syntactic parsing), biological sequence analysis (e.g., variant prediction), and emerging areas like graph neural networks and financial modeling. This work advances structured deep learning by fusing geometric and algebraic principles with attention mechanisms, offering a mathematically grounded alternative to data-driven models and paving the way for interpretable, efficient systems in complex domains.

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.

We develop new artificial neural network models for graded vector spaces, which are suitable when different features in the data have different significance (weights). This is the first time that such models are designed mathematically and they are expected to perform better than neural networks over usual vector spaces, which are the special case when the gradings are all 1s.