Sparse Variable Projection in Robotic Perception: Exploiting Separable Structure for Efficient Nonlinear Optimization

TL;DR

This paper introduces a variable projection method tailored for robotic perception problems with gauge symmetries, exploiting separability and sparsity to significantly accelerate nonlinear least-squares optimization.

Contribution

It develops a gauge-symmetry-aware VarPro scheme that constructs a matrix-free Schur complement operator for efficient nonlinear optimization in perception tasks.

Findings

Achieves 2x to 35x faster runtimes than existing methods.

Maintains high accuracy across SLAM, SNL, and SfM benchmarks.

Provides an open-source implementation and datasets.

Abstract

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.