Robotic perception often requires solving large nonlinear least-squares (NLS) problems. While sparsity has been widely exploited to scale solvers, a complementary and underused structure is \emph{separability}: some variables, such as visual landmarks, appear linearly in the residuals and admit a closed-form solution once the remaining variables, such as poses, are fixed. Variable projection (VarPro) exploits this structure by analytically eliminating the linear variables, yielding a reduced problem with favorable computational properties. However, its use in robotic perception has been limited by gauge symmetries, such as invariance to global translations and rotations, which introduce challenges for standard VarPro methods. We present SPARSER (\textbf{S}parsity \textbf{P}reserving \textbf{A}nalytic \textbf{R}eduction for \textbf{S}eparable \textbf{R}obotic \textbf{P}erception), a VarPro framework for gauge-symmetric problems that jointly exploits separability and sparsity. Our method constructs a \emph{matrix-free Schur complement operator} for efficient evaluation of reduced costs, gradients, and Hessian-vector products, enabling integration with iterative NLS solvers. We characterize the applicable problem class, identify common cases admitting further analytical simplifications, and show that IRLS-based robust costs preserve most of the exploitable structure. Across synthetic and real SLAM, SNL, and SfM benchmarks, SPARSER is on average 5×--7× faster than state-of-the-art baselines on CPU and GPU, with gains exceeding 40× on individual datasets. On outlier-corrupted multi-robot SLAM data, the robust variant is 2×--16× faster than a state-of-the-art GNC solver. We release open-source C++ code and all datasets.
Visual Place Recognition (VPR) in lifelong deployment requires continuous adaptation to new environments without catastrophic forgetting. In this paper, we propose SLAM, a Structured and Localized Analytic Manifold adaptation framework. Our framework elegantly unifies uncertainty-aware smoothing via Unscented transformation, topological space partitioning through a Gaussian Mixture Model (GMM), and H∞ robust bound optimization into a singular, unified closed-form analytical recursion. Exhaustive ablation studies demonstrate that while the synergistic combination of uncertainty smoothing and localized mapping (U+G configuration) achieves the state-of-the-art nominal accuracy of 27.5%, the full deployment of the H∞ bound does not require an architectural split; rather, it introduces a mathematically guaranteed minimax robust bound. This formulation enables the system to seamlessly modulate the intrinsic trade-off between nominal placement precision and worst-case disturbance attenuation through a single regularization parameter.
We present Caspar, a library that makes the power of modern GPUs more accessible in robotics and provides a state-of-the-art nonlinear GPU solver that can be applied to a wide range of different optimization problems. Caspar bridges the gap between expressive symbolic programming in Python and high-performance GPU runtimes in C++ by automatically generating optimized CUDA kernels from symbolic expressions. Building on the SymForce library, users can easily define and combine symbolic expressions, including Lie group operations, to generate custom CUDA kernels. To use Caspar as a solver, users need only define the symbolic residual functions; Caspar then uses symbolic differentiation to generate the necessary GPU kernels and interfaces to perform nonlinear optimization. In this paper, we present the core components of Caspar and showcase its performance by performing bundle adjustment on the Bundle Adjustment in the Large (BAL) dataset. We benchmark Caspar against other state-of-the-art bundle adjusters and show that it is 5 to 20 times faster than the best alternative, requires less memory, and achieves similar accuracy. This illustrates the benefit of our symbolic GPU programming approach. Caspar is released as part of SymForce and is freely available at https://github.com/symforce-org/symforce
A fundamental assumption in robotic perception is that the sensor's field of view (FoV) is fixed relative to the robot body. Motion-decoupled sensors, such as gimbal-mounted cameras and MEMS-based LiDARs, instead allow sensing direction to be controlled independently at runtime. This freedom creates a computational challenge: efficiently selecting useful viewing directions online in feature-dense environments. We propose PIVOT, a lightweight iterative method that optimizes sensor viewing direction along a fixed translation trajectory to maximize feature visibility. Under a conical FoV model, visibility depends only on the optical axis, yielding a two-degree-of-freedom optimization on the viewing sphere S2. Coordinate-free SO(3) exponential-map updates enable efficient continuous optimization without explicit angular parameterizations or exhaustive viewing-sphere search. Monte Carlo evaluations retain 98.1--99.6% of brute-force visibility with a 76--85x speedup. Photorealistic simulation and real-world experiments further demonstrate improved visual localization robustness and practical viewpoint control on a quadruped robot.