Distributional Stability of Tangent-Linearized Gaussian Inference on Smooth Manifolds

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 <inline-formula><tex-math notation="LaTeX">$W_{2}$</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">$(\mu,\Sigma)$</tex-math></inline-formula> and curvature/reach surrogates. Circle and planar-pushing experiments validate the predicted calibration transition near <inline-formula><tex-math notation="LaTeX">$\sqrt{\Vert \Sigma \Vert _{\text{op}}}/R\approx 1/6$</tex-math></inline-formula> 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.