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

arXiv: https://arxiv.org/abs/2512.07969 · HTML: https://mlsanity.com/paper/2512.07969/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 42 references that are in the map.
- First posted on arXiv 8 Dec 2025.

## Builds on (most cited first, 13 of 13)

- [Past, Present, and Future of Simultaneous Localization And Mapping: Towards the Robust-Perception Age](https://mlsanity.com/paper/1606.05830/) (2016)
- [SE-Sync: A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group](https://mlsanity.com/paper/1612.07386/) (2016)
- [Artificial Intelligence for Long-Term Robot Autonomy: A Survey](https://mlsanity.com/paper/1807.05196/) (2018)
- [Power Variable Projection for Initialization-Free Large-Scale Bundle Adjustment](https://mlsanity.com/paper/2405.05079/) (2024)
- [Building Rome with Convex Optimization](https://mlsanity.com/paper/2502.04640/) (2025)
- [Certifiably Correct Range-Aided SLAM](https://mlsanity.com/paper/2302.11614/) (2023)
- [Asynchronous and Parallel Distributed Pose Graph Optimization](https://mlsanity.com/paper/2003.03281/) (2020)
- [Majorization Minimization Methods for Distributed Pose Graph Optimization](https://mlsanity.com/paper/2108.00083/) (2021)
- [Benign landscapes of low-dimensional relaxations for orthogonal synchronization on general graphs](https://mlsanity.com/paper/2307.02941/) (2023)
- [SCORE: A Second-Order Conic Initialization for Range-Aided SLAM](https://mlsanity.com/paper/2210.03177/) (2022)
- [Large Scale Photometric Bundle Adjustment](https://mlsanity.com/paper/2008.11762/) (2020)
- [Sensor network localization has a benign landscape after low-dimensional relaxation](https://mlsanity.com/paper/2507.15662/) (2025)
- [Asynchronous Distributed Smoothing and Mapping via On-Manifold Consensus ADMM](https://mlsanity.com/paper/2310.12320/) (2023)

## Built on by (most cited first, 1 of 1)

- [Dr-BA: Separable Optimization for Direct Radar Bundle Adjustment & Localization](https://mlsanity.com/paper/2605.07041/) (2026)

## Abstract (arXiv)

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