Robotic perception often requires solving large nonlinear least-squares (NLS) problems. While sparsity has been well-exploited to scale solvers, a complementary and underexploited structure is \emph{separability} -- where some variables (e.g., visual landmarks) appear linearly in the residuals and, for any estimate of the remaining variables (e.g., poses), have a closed-form solution. Variable projection (VarPro) methods are a family of techniques that exploit this structure by analytically eliminating the linear variables and presenting a reduced problem in the remaining variables that has favorable properties. However, VarPro has seen limited use in robotic perception; a major challenge arises from gauge symmetries (e.g., cost invariance to global shifts and rotations), which are common in perception and induce specific computational challenges in standard VarPro approaches. We present a VarPro scheme designed for problems with gauge symmetries that jointly exploits separability and sparsity. Our method can be applied as a one-time preprocessing step to construct a \emph{matrix-free Schur complement operator}. This operator allows efficient evaluation of costs, gradients, and Hessian-vector products of the reduced problem and readily integrates with standard iterative NLS solvers. We provide precise conditions under which our method applies, and describe extensions when these conditions are only partially met. Across synthetic and real benchmarks in SLAM, SNL, and SfM, our approach achieves up to \textbf{2$\times$--35$\times$ faster runtimes} than state-of-the-art methods while maintaining accuracy. We release an open-source C++ implementation and all datasets from our experiments.

翻译：机器人感知通常需要求解大规模非线性最小二乘问题。尽管稀疏性已被广泛用于扩展求解器规模，但一种互补且尚未充分开发的结构是可分离性——即某些变量（如视觉路标）在残差中线性出现，并且对于其余变量（如位姿）的任何估计值，这些线性变量具有闭式解。变量投影方法是一类利用该结构的技术，通过解析消除线性变量，将问题简化为仅包含剩余变量的优化问题，该简化问题具有更优的性质。然而，变量投影在机器人感知中的应用有限；主要挑战源于规范对称性（如成本对全局平移和旋转的不变性），这在感知问题中普遍存在，并给标准变量投影方法带来特定的计算难题。我们提出了一种专为具有规范对称性的问题设计的变量投影方案，该方案同时利用了可分离性和稀疏性。我们的方法可作为一次性预处理步骤，构建一个无矩阵舒尔补算子。该算子能够高效计算简化问题的成本、梯度及海森矩阵-向量乘积，并可无缝集成到标准迭代非线性最小二乘求解器中。我们给出了该方法适用的精确条件，并描述了当这些条件仅部分满足时的扩展方案。在SLAM、SNL和SfM的合成与真实基准测试中，我们的方法在保持精度的同时，实现了比现有先进方法快2倍至35倍的运行速度。我们开源了C++实现代码及实验中的所有数据集。