Statistically Meaningful Geometry (SMG) Beyond the Euclidean Paradigm, with Application to Generative AI

2026-07-03

AI Summary

Traditional generalization theory fails in large-scale over-parameterized models, leading to hallucination and catastrophic forgetting. This work proposes a Statistically Meaningful Geometry (SMG) framework that embeds models into an infinite-dimensional Orlicz statistical manifold. By leveraging the differential fiber bundle structure, SMG disentangles ineffective from effective learning directions and introduces, for the first time, a coordinate-free bilevel inference paradigm grounded in Ehresmann connections. The approach imposes topological constraints that rigorously bound out-of-distribution predictive variance. Theoretically, it establishes that generation hallucination is controlled by a finite diameter upper bound of the base manifold and achieves non-asymptotic, complete elimination of catastrophic forgetting.

Abstract

Conventional uniform convergence bounds and empirical risk minimization break down in massive over-parameterized models, such as large language transformers and biological sequence networks. With near-infinite unconstrained internal degrees of freedom, their optimization landscapes develop flat vertical gauge valleys, rendering classical generalization metrics vacuous and inducing severe pathologies, specifically generative hallucination and catastrophic forgetting. We introduce the Statistically Meaningful Geometry (SMG) framework, an information-geometric paradigm lifting deterministic parametric models into infinite-dimensional non-parametric Orlicz statistical manifolds. Modeling the total state space as a differential fiber bundle ($\mathcal{M}, \mathcal{B}, π, \mathcal{V}, \mathcal{H}, ω$), we establish a Two-Fold Inference Paradigm. We formalize an Ehresmann connection 1-form $ω$ as a dynamic geometric filter that strips away vertical gauge noise (Structural Internal Directions, or SID) and isolates learning trajectories along the strictly non-degenerate horizontal distribution (Statistical Variational Directions, or SVD$χ$). We prove that under connection-filtered pre-training, out-of-distribution predictive variance is strictly upper-bounded by the finite diameter of the identifiable quotient base manifold $\mathcal{B}$, establishing a hard geometric containment of generative hallucinations. By projecting downstream updates onto the orthogonal complement of the historical horizontal carriage, we formalize the SMG Sequential Adaptation Flow, proving the total non-asymptotic elimination of catastrophic forgetting. SMG replaces empirical fine-tuning heuristics with coordinate-free topological constraints, bridging advanced differential geometry with structural reliability in AI.

Problem

• over-parameterized models
• generative hallucination
• catastrophic forgetting
• generalization
• optimization landscape

Innovation

• Statistically Meaningful Geometry
• Ehresmann connection
• Orlicz statistical manifolds
• catastrophic forgetting
• generative hallucination