Hierarchical Grading in Large Language Models

• URL: http://arxiv.org/abs/2607.22757v1
• Date: Thu, 23 Jul 2026 20:14:48 GMT
• Title: Hierarchical Grading in Large Language Models
• Abstract summary: We introduce Graded Large Language Models (GLLMs)<n>GLLMs are a framework that equips the representation space of a transformer with a grading.<n>The construction extends the theory of graded neural networks and graded transformers to autoregressive language models.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce Graded Large Language Models (GLLMs), an algebraic framework that equips the representation space of a transformer with a grading and propagates the induced weighted scalar action through embeddings, self-attention, and the training objective. The construction extends the theory of graded neural networks and graded transformers to autoregressive language models while preserving expressive power, asymptotic computational complexity, and inference cost. The governing geometric picture is that of geometric invariant theory. The benefit of a grading is expressed by a Kempf--Ness functional on the grading torus; the grades that improve upon the uniform architecture form an open convex cone whose membership is decided by a Hilbert--Mumford-type criterion pairing a grade direction against two measurable profiles of the target and the data; the optimal grades are the coincidence point of two moment maps, given in closed form; and the ordinary transformer appears as a semistable isotropic point on the boundary of the cone: one member of a larger graded family rather than a distinguished optimum. Separately, for level-stratified targets we prove a minimax separation between the graded prior and its absence: over all estimators the risks of the graded and uniform target classes separate throughout an explicit window of sample sizes, by a factor that decays exponentially in the number of levels under geometric stratification. Both profiles are estimable offline, so the optimal grades solve a convex program certified before training begins. Because the grading is absorbed into the learned parameters after training, every GLLM compiles to a standard transformer of identical architecture and inference complexity.

Related papers

• Hyper-Fold: Exploring the Expressive Limit of Sequence-Geometry Learning for Proteins via Hypergraph Modeling [55.36837305276515]
  We show that the complete bilinear operator over content-geometry outer products--the sufficient statistic of all second-order interactions--is the expressive ceiling.<n>We then introduce Hyper-Fold, a rank-K separable convolutional backbone approaching this ceiling at message-passing cost.
  arXiv  Detail & Related papers  (2026-08-29T11:43:25Z)
• Tropical Algebraic Geometry for Neuronal Representations: An Arakelov-Green Measure Based Descriptor for Graph Learning [14.563432689352084]
  We propose a training-free geometric prior based on tropical algebraic geometry.<n>We convert spatial trees into cyclic metric graphs suitable for embedding into the Tropical Jacobian.<n>We show that the discrete Arakelov-Green measure, computed in closed form via the graph Laplacian's generalized inverse, decomposes exactly into the intrinsic path metric minus the unquantized polarization distance on this cover.
  arXiv  Detail & Related papers  (2026-08-05T05:34:50Z)
• Restricted Dynamic Geometric Complexity: Path-Space Reduction and Möbius--Jacobi Response [0.0]
  We study restricted dynamic geometric complexity as a path-space value problem.<n>The main response result is global on a Hadamard state space.
  arXiv  Detail & Related papers  (2026-07-08T09:36:45Z)
• Statistically Meaningful Geometry (SMG) Beyond the Euclidean Paradigm, with Application to Generative AI [0.7766379063545538]
  We introduce the Statistically Meaningful Geometry (SMG) framework, an information-geometric paradigm lifting parametric models into infinite-dimensional minimization non-asymptotic statistics.<n>We prove that under connection-filtered pre-training, out-of-distribution predictive variance is strictly upper-bounded by the finite diameter of the identifiable base manifold $mathcalB$, establishing a hard geometric containment of generative hallucinations.
  arXiv  Detail & Related papers  (2026-07-03T13:46:31Z)
• D-Convexity: A Unified Differentiable Convex Shape Prior via Quasi-Concavity for Data-driven Image Segmentation [5.445483153964301]
  Convexity is a fundamental geometric prior that underlies many natural and man-made structures.<n>We propose a unified, threshold-free convexity prior based on the quasi-concavity of the network's output mask function u.<n>Our analysis unifies a wide spectrum of previous convex shape models, from discrete 1-0-1 line constraints to curvature or signed distance Laplacian based level-set priors.
  arXiv  Detail & Related papers  (2026-05-19T00:27:41Z)
• The Geometric Wall: Manifold Structure Predicts Layerwise Sparse Autoencoder Scaling Laws [13.054357482525505]
  We conduct the first cross-layer SAE scaling study, fitting and regressing on 68 layers of Gemma 2 2B and 9B.<n>We find that manifold geometry predicts the per-layer width exponent in both models.
  arXiv  Detail & Related papers  (2026-05-11T02:24:46Z)
• Layerwise LQR for Geometry-Aware Optimization of Deep Networks [38.84575015854741]
  We introduce Layerwise LQR, a framework for learning structured inverse preconditioners under a global layerwise optimal-control objective.<n>We derive a scalable relaxation that learns diagonal, (E-)Kronecker-factored, or other structured inverse preconditioners.<n> Experiments on ResNets and Transformers show that LLQR improves optimization dynamics and often translates these gains into improved final test performance.
  arXiv  Detail & Related papers  (2026-05-05T19:16:00Z)
• Graded Transformers [0.0]
  We introduce the Graded Transformer framework, a new class of sequence models that embed inductive biases through grading on vector spaces.<n>The framework enables adaptive feature prioritization, overcoming limitations of fixed grades in earlier models.<n>The Graded Transformer provides a mathematically principled approach to hierarchical learning and neuro-symbolic reasoning.
  arXiv  Detail & Related papers  (2025-07-27T02:34:08Z)
• Generalized Gradient Norm Clipping & Non-Euclidean $(L_0,L_1)$-Smoothness [51.302674884611335]
  This work introduces a hybrid non-Euclidean optimization method which generalizes norm clipping by combining steepest descent and conditional gradient approaches.<n>We discuss how to instantiate the algorithms for deep learning and demonstrate their properties on image classification and language modeling.
  arXiv  Detail & Related papers  (2025-06-02T17:34:29Z)
• What Improves the Generalization of Graph Transformers? A Theoretical Dive into the Self-attention and Positional Encoding [67.59552859593985]
  Graph Transformers, which incorporate self-attention and positional encoding, have emerged as a powerful architecture for various graph learning tasks. This paper introduces first theoretical investigation of a shallow Graph Transformer for semi-supervised classification.
  arXiv  Detail & Related papers  (2024-06-04T05:30:16Z)
• Convergence of Adam Under Relaxed Assumptions [72.24779199744954]
  We show that Adam converges to $epsilon$-stationary points with $O(epsilon-4)$ gradient complexity under far more realistic conditions. We also propose a variance-reduced version of Adam with an accelerated gradient complexity of $O(epsilon-3)$.
  arXiv  Detail & Related papers  (2023-04-27T06:27:37Z)
• Leveraging Non-uniformity in First-order Non-convex Optimization [93.6817946818977]
  Non-uniform refinement of objective functions leads to emphNon-uniform Smoothness (NS) and emphNon-uniform Lojasiewicz inequality (NL) New definitions inspire new geometry-aware first-order methods that converge to global optimality faster than the classical $Omega (1/t2)$ lower bounds.
  arXiv  Detail & Related papers  (2021-05-13T04:23:07Z)
• Self-supervised Geometric Perception [96.89966337518854]
  Self-supervised geometric perception is a framework to learn a feature descriptor for correspondence matching without any ground-truth geometric model labels. We show that SGP achieves state-of-the-art performance that is on-par or superior to the supervised oracles trained using ground-truth labels.
  arXiv  Detail & Related papers  (2021-03-04T15:34:43Z)
• Cogradient Descent for Bilinear Optimization [124.45816011848096]
  We introduce a Cogradient Descent algorithm (CoGD) to address the bilinear problem. We solve one variable by considering its coupling relationship with the other, leading to a synchronous gradient descent. Our algorithm is applied to solve problems with one variable under the sparsity constraint.
  arXiv  Detail & Related papers  (2020-06-16T13:41:54Z)
• Towards Better Understanding of Adaptive Gradient Algorithms in Generative Adversarial Nets [71.05306664267832]
  Adaptive algorithms perform gradient updates using the history of gradients and are ubiquitous in training deep neural networks. In this paper we analyze a variant of OptimisticOA algorithm for nonconcave minmax problems. Our experiments show that adaptive GAN non-adaptive gradient algorithms can be observed empirically.
  arXiv  Detail & Related papers  (2019-12-26T22:10:10Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.