# A Subspace-Constrained Tyler's Estimator and its Applications to Structure from Motion

arXiv: https://arxiv.org/abs/2404.11590 · HTML: https://mlsanity.com/paper/2404.11590/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 51 references that are in the map.
- First posted on arXiv 17 Apr 2024.

## Builds on (most cited first, 16 of 16)

- [Image Matching across Wide Baselines: From Paper to Practice](https://mlsanity.com/paper/2003.01587/) (2020)
- [MAGSAC: marginalizing sample consensus](https://mlsanity.com/paper/1803.07469/) (2018)
- [Resilience: A Criterion for Learning in the Presence of Arbitrary Outliers](https://mlsanity.com/paper/1703.04940/) (2017)
- [Robust Camera Location Estimation by Convex Programming](https://mlsanity.com/paper/1412.0165/) (2014)
- [Robustly Learning a Gaussian: Getting Optimal Error, Efficiently](https://mlsanity.com/paper/1704.03866/) (2017)
- [An Overview of Robust Subspace Recovery](https://mlsanity.com/paper/1803.01013/) (2018)
- [Fast, Robust and Non-convex Subspace Recovery](https://mlsanity.com/paper/1406.6145/) (2014)
- [A Well-Tempered Landscape for Non-convex Robust Subspace Recovery](https://mlsanity.com/paper/1706.03896/) (2017)
- [Algebraic Characterization of Essential Matrices and Their Averaging in Multiview Settings](https://mlsanity.com/paper/1904.02663/) (2019)
- [GPSfM: Global Projective SFM Using Algebraic Constraints on Multi-View Fundamental Matrices](https://mlsanity.com/paper/1812.00426/) (2018)
- [A New Rank Constraint on Multi-view Fundamental Matrices, and its Application to Camera Location Recovery](https://mlsanity.com/paper/1702.03023/) (2017)
- [Rigorous Guarantees for Tyler's M-estimator via quantum expansion](https://mlsanity.com/paper/2002.00071/) (2020)
- [Robust Subspace Recovery with Adversarial Outliers](https://mlsanity.com/paper/1904.03275/) (2019)
- [Scalable Cluster-Consistency Statistics for Robust Multi-Object Matching](https://mlsanity.com/paper/2201.04797/) (2022)
- [Stochastic and Private Nonconvex Outlier-Robust PCA](https://mlsanity.com/paper/2203.09276/) (2022)
- [Theoretical Guarantees for the Subspace-Constrained Tyler's Estimator](https://mlsanity.com/paper/2403.18658/) (2024)

## Built on by (most cited first, 8 of 8)

- [Tensor-Based Synchronization and the Low-Rankness of the Block Trifocal Tensor](https://mlsanity.com/paper/2409.09313/) (2024)
- [Cycle-Sync: Robust Global Camera Pose Estimation through Enhanced Cycle-Consistent Synchronization](https://mlsanity.com/paper/2511.02329/) (2025)
- [T-Rex: Fitting a Robust Factor Model via Expectation-Maximization](https://mlsanity.com/paper/2505.12117/) (2025)
- [Unsupervised Feature Selection via Robust Autoencoder and Adaptive Graph Learning](https://mlsanity.com/paper/2512.18720/) (2025)
- [QuadSync: Quadrifocal Tensor Synchronization via Tucker Decomposition](https://mlsanity.com/paper/2602.22639/) (2026)
- [TriDE: Triangle-Consistent Translation Directions for Global Camera Pose Estimation](https://mlsanity.com/paper/2605.06889/) (2026)
- [The Sharp Phase Transition of Tyler's M-Estimator for Robust Subspace Recovery](https://mlsanity.com/paper/2606.06782/) (2026)
- [Efficient Test-time Adaptation through Candidate Verification and Divergence Shifts](https://mlsanity.com/paper/2610.06147/) (2026)

## Abstract (arXiv)

We present the subspace-constrained Tyler's estimator (STE) designed for recovering a low-dimensional subspace within a dataset that may be highly corrupted with outliers. STE is a fusion of the Tyler's M-estimator (TME) and a variant of the fast median subspace. Our theoretical analysis suggests that, under a common inlier-outlier model, STE can effectively recover the underlying subspace, even when it contains a smaller fraction of inliers relative to other methods in the field of robust subspace recovery. We apply STE in the context of Structure from Motion (SfM) in two ways: for robust estimation of the fundamental matrix and for the removal of outlying cameras, enhancing the robustness of the SfM pipeline. Numerical experiments confirm the state-of-the-art performance of our method in these applications. This research makes significant contributions to the field of robust subspace recovery, particularly in the context of computer vision and 3D reconstruction.