The SMG-Yau-Yau Filter and Lossless Data Assimilation: Resolving Infinite Lie Algebras via Statistical Fiber Theory

2026-09-23

AI Summary

本文通过引入基于统计纤维理论的SMG-Yau-Yau滤波器和无损数据同化框架，解决了由于无限李代数导致的经典非线性滤波器发散问题。

Abstract

Classical continuous-time non-linear filtering fails in generic non-linear state spaces due to an infinite Lie algebraic derivative explosion ($\dim(\mathcal{E}) = \infty$), leading to filter divergence. To resolve this four-decade crisis, we introduce the SMG-Yau-Yau filter and lossless data assimilation framework built on Statistical Fiber Theory over an Orlicz manifold $\mathcal{M}$. By equipping $\mathcal{M}$ with a Riemannian submersion and an Ehresmann connection, the unconstrained Duncan-Mortensen-Zakai score velocity field is orthogonally decomposed into Statistically Verifiable Directions ($\text{SVD}χ_f$) and Structural Internal Directions ($\text{SID}_f$). System non-linearities and unclosed Lie commutators are orthogonally quarantined in $\text{SID}_f$, protecting macroscopic base parameters from spatial derivative pollution while preserving total score variance energy. We unify the asymptotics through a Dual-Axis Collapse Mechanism: proving our model identically recovers classical Yau-Yau dynamics when $\dim(\mathcal{E}) < \infty$, while large-sample limits ($N \to \infty$) induce a thermodynamic quench that flattens infinite-dimensional geometry under $\dim(\mathcal{E}) = \infty$. Finally, we formulate the Active Acausal Tension (AAT) functional to monitor accumulated model misspecification online. Exceeding a topological capacity threshold triggers Gauge Symmetry Breaking (GSB), which dynamically expands base coordinates ($d \to d+1$) to ensure non-asymptotic stability and convergence to the exact state density.

Problem

• non-linear filtering
• infinite Lie algebra
• filter divergence

Innovation

• SMG-Yau-Yau filter
• Lossless Data Assimilation
• Statistical Fiber Theory
• Active Acausal Tension (AAT)
• Gauge Symmetry Breaking (GSB)