Distributional Stability of Tangent-Linearized Gaussian Inference on Smooth Manifolds
Provides practical diagnostics for when to switch from single-chart linearization to more complex manifold inference methods in robotics.
The paper derives explicit non-asymptotic stability bounds for tangent-linearized Gaussian inference on smooth manifolds, showing that normal-direction uncertainty is the dominant failure mode when locality breaks, with a calibration transition near sqrt(||Σ||_op)/R ≈ 1/6.
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. Code and Jupyter notebooks are available at https://github.com/mikigom/StabilityTLGaussian.