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

2025-12-08

AI Summary

This work addresses large-scale nonlinear least-squares (NLS) problems in robotic perception, where canonical symmetries—such as global translation and rotation invariance—induce numerical instability and scalability bottlenecks in conventional variable projection (VarPro) methods. We propose the first symmetry-compatible sparse VarPro framework. Our method analytically eliminates linear variables (e.g., visual landmarks) while leveraging sparsity in the underlying factor graph and employing a matrix-free Schur complement operator, enabling seamless integration into standard iterative solvers. Key innovations include explicit modeling and nullspace suppression of symmetry-induced degeneracies, yielding closed-form variable projection and numerically stable Schur complement computation. Evaluated on SLAM, sensor network localization (SNL), and structure-from-motion (SfM) benchmarks, our approach achieves 2–35× speedup over baselines with no loss in accuracy. The implementation and experimental data are publicly available.

Technology Category

• Intelligent Robots: State Estimation
• Computer Vision: Visual Reasoning & Symbolic Representations
• Machine Learning: Neuro-Symbolic Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 extbf{2$ imes$--35$ imes$ faster runtimes} than state-of-the-art methods while maintaining accuracy. We release an open-source C++ implementation and all datasets from our experiments.

Problem

• Exploits separability and sparsity in robotic perception optimization
• Addresses gauge symmetry challenges in variable projection methods
• Enables faster nonlinear least-squares solving for SLAM and SfM

Innovation

• Variable projection exploiting separability and sparsity
• Matrix-free Schur complement operator for efficient computation
• Handles gauge symmetries in robotic perception problems