Distributional Stability of Tangent-Linearized Gaussian Inference on Smooth Manifolds

Supplementary notebooks for the IEEE Robotics and Automation Letters (RA-L) paper:

Distributional Stability of Tangent-Linearized Gaussian Inference on Smooth Manifolds

Junghoon Seo, Hakjin Lee, and Jaehoon Sim

The notebooks reproduce and extend the paper's circle and planar-pushing experiments for tangent-linearized Gaussian marginalization and conditioning on smooth manifolds.

Requirements

• Python 3.9+
• numpy, scipy, matplotlib
• Jupyter, VS Code, or another .ipynb execution environment

All notebooks seed NumPy random generation with 42 for deterministic runs.

Notebooks

1. notebook_circle_benchmark.ipynb - Circle Benchmark

Reproduces the 2-D circle benchmark from Section V-A of the paper and connects the observed calibration regimes to the marginalization bound in Theorem III.1.

• Setting: manifold M = {x in R^2 : ||x|| = R}, curvature kappa = 1/R, reach rho = R; ambient X ~ N(mu, Sigma) with mu = (R + delta, 0) and linearization point mu_tilde = (R, 0).
• Baseline: isotropic covariance with delta = 0.2, R in {0.5, 1, 2}, and sigma/R in [0.02, 1.2].
• Stress tests: anisotropic covariance with eta = sigma_n/sigma_t in {0.5, 1, 2, 4} and offset sweeps in delta/R.
• Generates high-DPI PNGs in notebook_figures/, including the circle metrics and generality figures used by the paper, plus supplementary visual and phase diagnostics.

2. notebook_conditioning_comparison.ipynb - Conditioning Comparison

Checks the conditioning behavior discussed in Section V-A and Theorem IV.1. The key paper claim is that tangent-chart conditioning degrades earlier than the corresponding marginalization approximation: around the sigma/R = 1/6 locality marker it already has less overconfidence headroom, and the variance-ratio boundary is reached at a smaller spread.

• Exact conditioning uses the surface-measure density on the circle, p(theta) proportional to exp(-0.5 (x(theta)-mu)^T Sigma^{-1} (x(theta)-mu)).
• Linearized conditioning samples a Gaussian on the tangent line and retracts it to the circle.
• Experiments compare density shape, variance ratio, coverage, mode shift under tangential offset, chart-distortion terms, and the sigma/R = 0.5 moderate spread regime where density-shape artifacts become visible.

3. notebook_planar_pushing.ipynb - Planar Pushing

Reproduces the contact-rich planar-pushing experiment adapted from Guo et al. (ICRA 2025) and Qadri et al. (InCOpt, IROS 2022).

• Setting: rectangular box with (w, h) = (0.2, 1.0) m, circular probe radius r_p = 0.1 m, and n_steps = 50.
• Odometry noise: Q = diag(0.03^2, 0.03^2, 0.01^2).
• Computes constrained covariance by tangent-space conditioning, Sigma_con = N (N^T F N)^{-1} N^T.
• Reports trajectory-wide locality diagnostics, Monte Carlo covariance mismatch, and directional noise stress tests separating normal and tangential uncertainty.

Running

From this directory:

jupyter nbconvert --to notebook --execute notebook_circle_benchmark.ipynb
jupyter nbconvert --to notebook --execute notebook_conditioning_comparison.ipynb
jupyter nbconvert --to notebook --execute notebook_planar_pushing.ipynb

or open each notebook in Jupyter/VS Code and run all cells. The circle notebook writes figure files to notebook_figures/; the conditioning and planar-pushing notebooks render their figures inline for inspection or export.