Distributional Stability of Tangent-Linearized Gaussian Inference on Smooth Manifolds

2026-02-22

AI Summary

This work investigates the stability of Gaussian inference on smooth manifolds, where marginalization and conditioning typically yield non-Gaussian distributions influenced by underlying geometry, complicating the assessment of linearization-based methods. Focusing on tangent-space linearization, the study establishes the first explicit non-asymptotic Wasserstein-2 (W₂) stability bound, cleanly separating local second-order geometric distortion from non-local tail leakage effects. The proposed closed-form diagnostic depends only on the mean, covariance, and proxies for curvature or injectivity radius, revealing that normal-direction uncertainty dominates error when locality assumptions break down. Experiments on toroidal and planar systems demonstrate a sharp degradation in calibration performance when √|Σ|_op/R ≈ 1/6, providing a practical trigger for switching to multi-chart or sampling-based manifold inference schemes.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Security and Privacy: Large-scale security measurements

Abstract

Gaussian inference on smooth manifolds is central to robotics, but exact marginalization and conditioning are generally non-Gaussian and geometry-dependent. We study tangent-linearized Gaussian inference and derive explicit non-asymptotic $W_2$ stability bounds for projection marginalization and surface-measure conditioning. The bounds separate local second-order geometric distortion from nonlocal tail leakage and, for Gaussian inputs, yield closed-form diagnostics from $(μ,Σ)$ and curvature/reach surrogates. Circle and planar-pushing experiments validate the predicted calibration transition near $\sqrt{\|Σ\|_{\mathrm{op}}}/R\approx 1/6$ and indicate that normal-direction uncertainty is the dominant failure mode when locality breaks. These diagnostics provide practical triggers for switching from single-chart linearization to multi-chart or sample-based manifold inference.

Problem

• Gaussian inference
• smooth manifolds
• distributional stability
• tangent-linearization
• geometric distortion

Innovation

• tangent-linearized Gaussian inference
• Wasserstein stability
• manifold geometry
• non-asymptotic bounds
• projection marginalization