Robotic perception method speeds up calculations using variable projection
SPARSER: Sparse Variable Projection by Exploiting Separable Structure in Robotic Perception
Robotics
Summary
Robotic systems that see and understand environments often need to solve big math problems to figure out their surroundings. The authors developed a faster way to solve these problems by focusing on the parts that can be separated and simplified, especially when some parts affect the solution linearly. Their approach also handles tricky mathematical symmetries that confuse earlier methods. They tested their method on various robotics tasks and showed it can be five to seven times faster on average than current top solutions.
What this means in practice
- •For robotics engineers: Solve large-scale mapping and localization problems faster by exploiting separable problem structure in robotic perception.
- •For computer vision developers: Speed up 3D reconstruction tasks that combine poses and visual features by integrating a sparsity-aware variable projection solver.
Authors
Nikolas R. Sanderson, Andrew Fishberg, Haoyu Han, Heng Yang, Jonathan P. How, Hanumant Singh, Michael Everett, Alan Papalia
Abstract
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 $5\times$--$7\times$ faster than state-of-the-art baselines on CPU and GPU, with gains exceeding $40\times$ on individual datasets. On outlier-corrupted multi-robot SLAM data, the robust variant is $2\times$--$16\times$ faster than a state-of-the-art GNC solver. We release open-source C++ code and all datasets.