提出几何稳定分析方法，判断高斯推理在流形上的可靠性边界。
Distributional Stability of Tangent-Linearized Gaussian Inference on Smooth Manifolds
- 通过切线线性化分析，建立非渐近的稳定性界限。
- 发现当√‖Σ‖ₐₚ / R ≈ 1/6 时出现校准突变，误差显著上升。
- 提供可计算的诊断指标，指导从线性化切换到多图集推理。
光滑流形上的高斯推断是机器人学的核心问题，但精确边缘化与条件化通常非高斯且依赖几何结构。本文研究切线线性化高斯推断，推导出投影边缘化与曲面测度条件化的显式非渐近 $W_2$ 稳定性界。该界将局部二阶几何失真与非局部尾部泄漏分离，在高斯输入下，可由 $(μ,Σ)$ 与曲率/可达性代理变量得出闭式诊断。圆环与平面推动实验验证了在 √‖Σ‖ₐₚ / R ≈ 1/6 附近预测的校准转变，并表明当局部性失效时，法向不确定性为最主要失效模式。这些诊断提供了从单图集线性化切换至多图集或基于采样的流形推断的实际触发条件。代码与 Jupyter 笔记本见 https://github.com/mikigom/StabilityTLGaussian。
原文摘要 · Abstract (English)
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.
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