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Distributional Stability of Tangent-Linearized Gaussian Inference on Smooth Manifolds

Junghoon Seo, Hakjin Lee, Jaehoon Sim

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.

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 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 and curvature/reach surrogates. Circle and planar-pushing experiments validate the predicted calibration transition near 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.
Paper Structure (31 sections, 6 theorems, 77 equations, 6 figures, 1 table)

This paper contains 31 sections, 6 theorems, 77 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wasserstein distance
  4. Embedded submanifolds, curvature, and reach
  5. Retractions and charts
  6. Marginalization and conditioning onto manifolds
  7. Linearization principle for smooth manifolds
  8. W2 stability for Gaussian marginalization
  9. Geometric locality
  10. Gaussian specialization
  11. Interpretation and numerical illustration
  12. W2 stability for Gaussian conditioning
  13. Tangent-plane reference law
  14. Chart model and approximation
  15. Tail quantities
  16. ...and 16 more sections

Key Result

Theorem III.1

There exists a constant $C_{\mathrm{loc}}>0$ (depending only on dimensionless tube/curvature margins, e.g. $r/\rho$ and $r\kappa$, and on the ambient dimension) such that, with $A:=\{\|X-\tilde{\mu}\|\le r\}$ and $\varepsilon:=\mathbb{P}(A^c)$, where

Figures (6)

  • Figure 1: $W_2$ stability of marginalizing a Gaussian onto a circular arc. Exact metric projection $g$ vs. the tangent--retraction approximation $\hat{g}$.
  • Figure 2: $W_2$ stability of conditioning a Gaussian onto a circular arc. Surface-measure conditioning vs. a tangent-plane chart approximation.
  • Figure 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 $R\in\{0.5,1,2\}$.
  • Figure 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 $\delta/R$.
  • Figure 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 $(\varrho_k^{\mathrm{MC}},\Delta_k)$.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Theorem III.1: W$_2$ stability of marginalization under tangent--retraction
  • Remark III.2: Practical calibration of $C_{\mathrm{loc}}$
  • proof : Proof of Theorem \ref{['thm:W2-marg']}
  • Lemma III.3: Local quadratic comparison of projection and tangent--retraction
  • proof
  • Theorem IV.1: W$_2$ stability of surface-measure conditioning under chart linearization
  • Lemma IV.2: Truncation bound
  • proof
  • Lemma IV.3: Diameter--TV control of $W_2$ on bounded support
  • proof
  • ...and 3 more