Jul 3, 2026 · cs.LGJ/K move · Enter open · S save
Bing Cheng, Yi-Shuai Niu, Howell Tong, Shing-Tung Yau
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China · AMSS Center for Forecasting Science, Chinese Academy of Sciences, Beijing, China · Beijing Institute of Mathematical Sciences and Applications (BIMSA), Beijing, China · Department of Statistics, London School of Economics and Political Science, London WC2A 2AE, UK · Department of Statistics and Data Science, Tsinghua University, Beijing 100084, China
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 (
M,B,π,V,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
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.