Distributional Stability of Tangent-Linearized Gaussian Inference on Smooth Manifolds

Junghoon Seo; Hakjin Lee; Jaehoon Sim

cs.ROFeb 22, 2026

Distributional Stability of Tangent-Linearized Gaussian Inference on Smooth Manifolds

Authors: Junghoon Seo, Hakjin Lee, Jaehoon Sim

Organizations: AI Robot Team, PIT IN Corp., South Korea.

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. Code and Jupyter notebooks are available at https://github.com/mikigom/StabilityTLGaussian.

Figures & tables

Fig. 1 : $W_{2}$ stability of marginalizing a Gaussian onto a circular arc. Exact metric projection $g$ vs. the tangent–retraction approximation $\hat{g}$ .

Fig. 2 :

W_{2}

stability of conditioning a Gaussian onto a circular arc. Surface-measure conditioning vs. a tangent-plane chart approximation.

Fig. 3 : Circle sweep ( $\delta=0.2$ ) over $\sigma/R$ : (a) variance ratio $\varrho=\mathrm{Var}_{\mathrm{lin}}/\mathrm{Var}_{\mathrm{exact}}$ , (b) realized $95\%$ coverage, and (c) normalized diagnostics on $\sqrt{\|\Sigma\|_{\mathrm{op}}}/R$ : tightened theoretical $W_{2}$ bound (data-calibrated $C_{\mathrm{tail}}$ with strict upper-envelope safeguard) and empirical $W_{2}$ proxy. Dashed line marks $\sigma/R=1/6$ . Colors indicate $R\in\{0.5,1,2\}$ .

Fig. 4 : Circle generality stress tests: (a) anisotropic sweep ( $\eta=\sigma_{n}/\sigma_{t}$ ) versus $\sqrt{\|\Sigma\|_{\mathrm{op}}}/R$ , (b) tail proxy $\varepsilon$ versus offset $\delta/R$ , and (c) coverage degradation versus $\delta/R$ .

	$\mathrm{tr}(\Sigma_{xy})$		$\lambda_{\max}(\Sigma_{xy})$		Red.	$\\|\Delta\Sigma_{xy}\\|_{F}$
$k$	Unc.	Con.	Unc.	Con.	(%)
1	0.0018	0.0009	0.0009	0.0009	50.5	0.0009
13	0.0234	0.0105	0.0117	0.0105	55.3	0.0118
25	0.0450	0.0190	0.0225	0.0190	57.8	0.0228
37	0.0666	0.0278	0.0333	0.0278	58.2	0.0337
49	0.0882	0.0377	0.0441	0.0377	57.3	0.0446

TABLE I : Quantitative covariance metrics for planar pushing at timesteps

Fig. 5 : Trajectory-wide planar-pushing diagnostics: (a) curvature/reach proxies $(\hat{\kappa}_{k},\hat{\rho}_{k})$ , (b) spread $s_{k}$ and locality indicator $s_{k}/\hat{\rho}_{k}$ , (c) Monte Carlo mismatch $(\varrho_{k}^{\mathrm{MC}},\Delta_{k})$ .

Fig. 6 : Directional scaling at the terminal planar-pushing pose: isotropic, normal-only, and tangential-only covariance inflation.

Bayesian Inference

Bayesian inference updates probability distributions over unknowns with evidence to quantify uncertainty and guide prediction and decisions.

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Division of Mathematical Sciences, Nanyang Technological University. · School of Statistics and Data Science, Shanghai University of Finance and Economics.

Nonparametric Regression

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We consider nonparametric tangent vector field regression on a Riemannian manifold without boundary. Because responses at different points lie in different tangent spaces, the proposed kernel estimator first parallel transports nearby responses to the target tangent space and then forms a volume-corrected local average. We first derive its uniform second-order bias, finite-bandwidth covariance, and stochastic rate. For simultaneous inference, the tangent norm is written as a supremum over the unit tangent bundle. Exact covariance whitening gives a unit-variance Gaussian field whose correlation length is of order

h

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Abstract

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