# Learning to Solve Hard Minimal Problems

arXiv: https://arxiv.org/abs/2112.03424 · HTML: https://mlsanity.com/paper/2112.03424/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 74 references that are in the map.
- First posted on arXiv 6 Dec 2021.

## Builds on (most cited first, 12 of 12)

- [Image Matching across Wide Baselines: From Paper to Practice](https://mlsanity.com/paper/2003.01587/) (2020)
- [Neighbourhood Consensus Networks](https://mlsanity.com/paper/1810.10510/) (2018)
- [Beyond Gr"obner Bases: Basis Selection for Minimal Solvers](https://mlsanity.com/paper/1803.04360/) (2018)
- [PLMP -- Point-Line Minimal Problems in Complete Multi-View Visibility](https://mlsanity.com/paper/1903.10008/) (2019)
- [PL1P -- Point-line Minimal Problems under Partial Visibility in Three Views](https://mlsanity.com/paper/2003.05015/) (2020)
- [Minimal Problems for the Calibrated Trifocal Variety](https://mlsanity.com/paper/1611.05947/) (2016)
- [A Minimal Solution for Two-view Focal-length Estimation using Two Affine Correspondences](https://mlsanity.com/paper/1706.01649/) (2017)
- [Galois/monodromy groups for decomposing minimal problems in 3D reconstruction](https://mlsanity.com/paper/2105.04460/) (2021)
- [Five-point Fundamental Matrix Estimation for Uncalibrated Cameras](https://mlsanity.com/paper/1803.00260/) (2018)
- [A sparse resultant based method for efficient minimal solvers](https://mlsanity.com/paper/1912.10268/) (2019)
- [A Minimal Closed-Form Solution for Multi-Perspective Pose Estimation using Points and Lines](https://mlsanity.com/paper/1807.09970/) (2018)
- [On the Existence of Epipolar Matrices](https://mlsanity.com/paper/1510.01401/) (2015)

## Built on by (most cited first, 19 of 19)

- [Continuation Path Learning for Homotopy Optimization](https://mlsanity.com/paper/2307.12551/) (2023)
- [Snapshot of Algebraic Vision](https://mlsanity.com/paper/2210.11443/) (2022)
- [Level-S2fM: Structure from Motion on Neural Level Set of Implicit Surfaces](https://mlsanity.com/paper/2211.12018/) (2022)
- [GlobalFlowNet: Video Stabilization using Deep Distilled Global Motion Estimates](https://mlsanity.com/paper/2210.13769/) (2022)
- [Handbook on Leveraging Lines for Two-View Relative Pose Estimation](https://mlsanity.com/paper/2309.16040/) (2023)
- [On the Learn-to-Optimize Capabilities of Transformers in In-Context Sparse Recovery](https://mlsanity.com/paper/2410.13981/) (2024)
- [Ground Truth Inference for Weakly Supervised Entity Matching](https://mlsanity.com/paper/2211.06975/) (2022)
- [Order-One Rolling Shutter Cameras](https://mlsanity.com/paper/2403.11295/) (2024)
- [An Accurate and Real-time Relative Pose Estimation from Triple Point-line Images by Decoupling Rotation and Translation](https://mlsanity.com/paper/2403.11639/) (2024)
- [Minimal Perspective Autocalibration](https://mlsanity.com/paper/2405.05605/) (2024)
- [Trifocal Tensor and Relative Pose Estimation with Known Vertical Direction](https://mlsanity.com/paper/2512.19110/) (2025)
- [Neural Predictor-Corrector: Solving Homotopy Problems with Reinforcement Learning](https://mlsanity.com/paper/2602.03086/) (2026)
- [Attention is All They Need: Exploring the Media Archaeology of the Computer Vision Research Paper](https://mlsanity.com/paper/2209.11200/) (2022)
- [Practical solutions to the relative pose of three calibrated cameras](https://mlsanity.com/paper/2303.16078/) (2023)
- [Simulator HC: Regression-based Online Simulation of Starting Problem-Solution Pairs for Homotopy Continuation in Geometric Vision](https://mlsanity.com/paper/2411.03745/) (2024)
- [Three-view Focal Length Recovery From Homographies](https://mlsanity.com/paper/2501.07499/) (2025)
- [Differential pose optimization in descriptor space -- Combining Geometric and Photometric Methods for Motion Estimation](https://mlsanity.com/paper/2602.14297/) (2026)
- [Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices](https://mlsanity.com/paper/2602.23450/) (2026)
- [Global Convergence of Sampling-Based Nonconvex Optimization through Diffusion-Style Smoothing](https://mlsanity.com/paper/2605.16520/) (2026)

## Abstract (arXiv)

We present an approach to solving hard geometric optimization problems in the RANSAC framework. The hard minimal problems arise from relaxing the original geometric optimization problem into a minimal problem with many spurious solutions. Our approach avoids computing large numbers of spurious solutions. We design a learning strategy for selecting a starting problem-solution pair that can be numerically continued to the problem and the solution of interest. We demonstrate our approach by developing a RANSAC solver for the problem of computing the relative pose of three calibrated cameras, via a minimal relaxation using four points in each view. On average, we can solve a single problem in under 70 μ s. We also benchmark and study our engineering choices on the very familiar problem of computing the relative pose of two calibrated cameras, via the minimal case of five points in two views.