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.

翻译：新型SE(3)估计目标的快速原型化在实践中仍存在困难。现代李群框架（GTSAM、manif、Sophus、SymForce、Ceres）通过不同的代码生成和自动微分策略面向一阶工作负载，每种策略均在手工推导的几何体与通用微分之间的特定接合处进行了优化。现存的关键缺口在于：从这些一阶原语出发，需要一种紧凑且支持自动微分安全的路径，以获取精确的海森矩阵、观测信息矩阵及高阶导数张量——这些正是实现精确牛顿步长、观测信息协方差估计与协方差修正所需的量。本文提出一种面向SE(3)负对数似然的解析/自动微分混合方案。实践者仅需编写一次对标量类型泛化的负对数似然梯度，并在点作用接口y=Tx处设置解析/自动微分的接合点：该接口前使用闭式李群雅可比矩阵，接口后仅应用自动微分。同一源码随后可实例化为：普通浮点标量（用于梯度）、向量播种对偶数（通过单次前向模式传递获取精确海森矩阵），以及嵌套对偶数（用于高阶导数张量）。基于代表性6自由度5地标SE(3)负对数似然基准测试，本文倡导的播种海森路径比该自动微分梯度的有限差分法快约5倍，且与嵌套自动微分预言机精确匹配至机器精度。该实现相对于纯自动微分基线仅增加约70行解析雅可比代码。我们还识别并修正了标准SO(3)/SE(3)标量基中的可移除奇点（该奇点会在播种自动微分下于原点处产生NaN），并审计了需要该稳定化基的李群导数张量。最终成果为：从快速编写的SE(3)目标到精确高阶导数提供了一条实用路径，具备可预测运行时且无需调整有限差分参数。