Distributional Stability of Tangent-Linearized Gaussian Inference on Smooth Manifolds

1. Problem Statement

In robotics, probabilistic inference often occurs on smooth manifolds (e.g., Lie groups like $SO(3)$, $SE(3)$, or contact constraint sets) where state spaces are intrinsically nonlinear. Standard estimation pipelines typically employ a "linearize–infer–retract" workflow:

1. Linearize: Approximate the manifold locally with its tangent space at a point $\tilde{\mu}$.
2. Infer: Perform Gaussian marginalization or conditioning on the tangent space (affine subspace) using closed-form identities.
3. Retract: Map the resulting distribution back to the manifold.

The Core Issue: While this approach is computationally efficient, it is unclear when it is reliable. Exact marginalization and conditioning on curved manifolds generally yield non-Gaussian distributions. The paper addresses the lack of explicit, non-asymptotic error bounds that quantify the discrepancy between the exact manifold-induced law and the tangent-linearized approximation. Specifically, it seeks to determine how curvature, covariance scale, and mean offset affect the validity of single-chart linearization.

2. Methodology

The authors formalize reliability as a distributional stability problem, quantifying the discrepancy between the exact distribution and the approximation using the 2-Wasserstein distance ($W_2$). The $W_2$ metric is chosen because it metrizes weak convergence with second moments, directly controlling errors in mean and covariance.

The study analyzes two canonical operations:

• Projection Marginalization: Pushing an ambient Gaussian $X \sim \mathcal{N}(\mu, \Sigma)$ onto the manifold $M$ via the metric projection $g: \mathbb{R}^n \to M$.
• Surface-Measure Conditioning: Restricting the ambient Gaussian density to $M$ and renormalizing with respect to the surface measure (volume form) of $M$.

Theoretical Framework:
The authors derive error bounds by decomposing the error into two components:

1. Local Geometric Distortion: The error arising from the second-order mismatch between the manifold's curvature and the flat tangent space within a local neighborhood.
2. Nonlocal Tail Leakage: The error caused by probability mass "leaking" outside the local neighborhood where the linearization is valid.

Key geometric concepts used include:

• Reach ($\rho$): The largest radius such that every point within that distance has a unique nearest point on the manifold.
• Curvature ($\kappa$): Bounded by the operator norm of the second fundamental form.
• Retraction/Charts: Mappings from the tangent space back to the manifold with quadratic accuracy bounds.

3. Key Contributions

A. Explicit Non-Asymptotic $W_2$ Bounds

The paper derives rigorous upper bounds for the $W_2$ distance between the exact and approximate distributions:

• For Marginalization (Theorem III.1): The bound separates local error (scaling with curvature $\kappa$ and retraction error $\kappa_R$) from tail leakage (controlled by the probability mass outside the localization radius).
  $$W_2 \leq C_{loc}(\kappa + \kappa_R)\sqrt{E[\|X-\tilde{\mu}\|^4]} + C_{tail}\epsilon^{1/4}$$
• For Conditioning (Theorem IV.1): The bound accounts for density shape distortions (Jacobian factors and log-density differences) in addition to mass transport. It introduces a "curvature/volume mismatch exponent" $\eta_r$ that quantifies how constraint curvature and prior precision warp the tangent approximation.

B. Closed-Form Diagnostics for Gaussian Inputs

For Gaussian inputs, the authors specialize the bounds to provide computable diagnostics based on the mean ($\mu$), covariance ($\Sigma$), and local geometry ($\rho, \kappa$):

• Stability Metric ($k_\rho$): A key diagnostic is the ratio $k_\rho = \sqrt{\|\Sigma\|_{op}} / \rho$, representing the ratio of the uncertainty spread to the manifold's reach.
• Offset Metric ($s_\delta$): The ratio of the mean offset from the linearization point to the reach, $\|\delta\|/\rho$.
• Failure Prediction: The bounds predict that linearization becomes unreliable when $k_\rho \gtrsim 1/6$ or when the mean offset is large.

C. Identification of Failure Modes

The analysis reveals that normal-direction uncertainty is the dominant failure mode. Uncertainty aligned with the manifold's normal vector stresses curvature and reach the most, leading to rapid calibration degradation compared to tangential uncertainty.

4. Experimental Results

The authors validate their theoretical predictions using two benchmarks:

A. 2-D Circle Benchmark

• Setup: A Gaussian is projected/conditioned onto a circle of radius $R$.
• Findings:
  • Transition Point: The calibration transition (where the linear approximation becomes overconfident) occurs consistently at $\sqrt{\|\Sigma\|_{op}}/R \approx 1/6$.
  • Anisotropy: The spectral norm $\|\Sigma\|_{op}$ serves as an effective diagnostic even for anisotropic covariances.
  • Offset Sensitivity: Large mean offsets ($\delta$) cause tail leakage and coverage degradation even if curvature is low.
  • Conditioning vs. Marginalization: Conditioning is more fragile than marginalization; it breaks down at a smaller spread ($\approx 0.30$ vs. $\approx 0.38$) due to density-shape artifacts (Jacobian distortion) that $W_2$ may underestimate.

B. Planar Pushing (Contact-Rich Task)

• Setup: A robot pushes a box with hard contact constraints (manifold constraints).
• Findings:
  • Trajectory Analysis: The mismatch between linearized and Monte Carlo covariances correlates strongly with the locality indicator $s_k/\hat{\rho}_k$.
  • Directional Stress: Inflation of uncertainty in the normal direction causes significantly larger calibration errors than tangential inflation, confirming the theoretical prediction that normal uncertainty is the critical failure mode.
  • Practical Thresholds: For typical contact radii ($R_c \approx 0.1$ m), the threshold $\sqrt{\|\Sigma\|_{op}} \lesssim 17$ mm is easily violated by process noise accumulation.

5. Significance and Practical Implications

• Runtime Calibration Gate: The derived diagnostics ($k_\rho$ and $s_\delta$) provide a practical, low-cost trigger for robotics solvers (e.g., GTSAM, iSAM2).
  • If $k_\rho \lesssim 1/6$ and $s_\delta \ll 1$: The single-chart tangent linearization is reliable.
  • If $k_\rho$ is large: Switch to iterated relinearization.
  • If $s_\delta$ is large: Immediate relinearization is required (e.g., after loop closures).
  • If both are large: Switch to multi-chart or particle-based inference.
• Asymmetry in Conditioning: The paper highlights that conditioning is structurally more sensitive than marginalization due to normalization constants amplifying density shape errors. This suggests that conditioning operations require stricter checks (e.g., coverage or KL-divergence checks) even when $W_2$ bounds appear moderate.
• Theoretical Foundation: This work bridges the gap between heuristic robotics estimation and rigorous geometric probability, providing the first non-asymptotic bounds that separate geometric distortion from tail leakage for Gaussian inference on manifolds.

In summary, the paper provides a mathematically rigorous framework to determine when tangent-linearized Gaussian inference is safe to use in robotics and when more complex, non-linear methods are necessary, preventing the use of overconfident and inaccurate uncertainty estimates in critical control and planning tasks.