提出SE(3)高阶导数的混合解析与自动微分方法，实现精确牛顿优化。
Exact Higher-Order Derivatives for SE(3) via Analytical/AD Methods
- 在点作用接口处用解析雅可比，之后用自动微分计算高阶导数。
- 单次前向传播即可获得精确海森矩阵，速度比有限差分快5倍。
- 适用于需要精确协方差估计的6自由度位姿优化任务。
现有李群框架多聚焦一阶导数计算，难以高效获取精确的海森矩阵、观测信息矩阵等高阶导数。本文提出一种混合解析/自动微分方法，对SE(3)负对数似然函数，在点作用接口 y = Tx 处使用闭式李群雅可比，其后仅用自动微分。同一源码通过普通浮点数求梯度，向量种子对偶数求海森矩阵，嵌套对偶数求更高阶张量。在典型6自由度、5特征点的SE(3) NLL任务中，该方法比有限差分法快约5倍，且精度达机器级。实现仅增加约70行解析雅可比代码。修复了标准SO(3)/SE(3)标量基在原点处的可移除奇点问题，避免对偶数微分产生NaN。结果提供了一条从快速编写的目标函数到精确高阶导数的可靠路径，运行时间可预测，无需有限差分调参。
原文摘要 · Abstract (English)
Fast prototyping of new SE(3) estimation objectives remains awkward in practice. Modern Lie-group frameworks -- GTSAM, manif, Sophus, SymForce, Ceres -- target first-order workloads through different code-generation and automatic-differentiation strategies, each optimized for a particular seam between hand-derived geometry and generic differentiation. The remaining gap is a compact, AD-safe path from these first-order primitives to exact Hessians, observed-information matrices, and higher-order derivative tensors: the quantities needed for exact Newton steps, observed-information covariance estimates, and covariance correction. This paper presents a hybrid analytical/AD recipe for SE(3) negative log-likelihoods. The practitioner writes the NLL gradient once, generic over a scalar type, and places the analytical/AD seam at the point-action interface y = Tx. Closed-form Lie-group Jacobians are used up to this interface; AD is applied only beyond it. The same source is then instantiated with ordinary floating-point scalars for gradients, vector-seeded dual numbers for exact Hessians in a single forward-mode pass, and nested dual numbers for higher-order derivative tensors. On a representative 6-DoF, 5-landmark SE(3) NLL, the advocated seeded-Hessian path is approximately 5x faster than finite-differencing the AD gradient on this benchmark while matching a nested-AD oracle to machine precision. The implementation adds roughly 70 lines of analytical-Jacobian code over an AD-only baseline. We also identify and fix a removable singularity in the standard SO(3)/SE(3) scalar basis that would otherwise produce NaNs at the origin under seeded AD, and we audit which Lie-group derivative tensors require this stabilized basis. The result is a practical path from rapidly written SE(3) objectives to exact higher-order derivatives, with predictable runtime and no finite-difference tuning.
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