# Eigendecomposition-free Training of Deep Networks with Zero Eigenvalue-based Losses

arXiv: https://arxiv.org/abs/1803.08071 · HTML: https://mlsanity.com/paper/1803.08071/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Mar 2018.

## Built on by (most cited first, 15 of 15)

- [MultiBodySync: Multi-Body Segmentation and Motion Estimation via 3D Scan Synchronization](https://mlsanity.com/paper/2101.06605/) (2021)
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- [SE(3) Diffusion Model-based Point Cloud Registration for Robust 6D Object Pose Estimation](https://mlsanity.com/paper/2310.17359/) (2023)
- [An Information-Theoretic Perspective on Variance-Invariance-Covariance Regularization](https://mlsanity.com/paper/2303.00633/) (2023)
- [Scribble-Supervised Semantic Segmentation by Uncertainty Reduction on Neural Representation and Self-Supervision on Neural Eigenspace](https://mlsanity.com/paper/2102.09896/) (2021)
- [Robust Differentiable SVD](https://mlsanity.com/paper/2104.03821/) (2021)
- [Fast Differentiable Matrix Square Root and Inverse Square Root](https://mlsanity.com/paper/2201.12543/) (2022)
- [Fast Differentiable Matrix Square Root](https://mlsanity.com/paper/2201.08663/) (2022)
- [MatchNorm: Learning-based Point Cloud Registration for 6D Object Pose Estimation in the Real World](https://mlsanity.com/paper/2203.15309/) (2022)
- [Real-time Trajectory-based Social Group Detection](https://mlsanity.com/paper/2304.05678/) (2023)
- [FroSSL: Frobenius Norm Minimization for Efficient Multiview Self-Supervised Learning](https://mlsanity.com/paper/2310.02903/) (2023)
- [End-to-end learning of keypoint detection and matching for relative pose estimation](https://mlsanity.com/paper/2104.01085/) (2021)
- [MinBackProp -- Backpropagating through Minimal Solvers](https://mlsanity.com/paper/2404.17993/) (2024)
- [Gaussian-Mixture-Model Q-Functions for Policy Iteration in Reinforcement Learning](https://mlsanity.com/paper/2512.18763/) (2025)
- [LoCUS: Learning Multiscale 3D-consistent Features from Posed Images](https://mlsanity.com/paper/2310.01095/) (2023)

## Abstract (arXiv)

Many classical Computer Vision problems, such as essential matrix computation and pose estimation from 3D to 2D correspondences, can be solved by finding the eigenvector corresponding to the smallest, or zero, eigenvalue of a matrix representing a linear system. Incorporating this in deep learning frameworks would allow us to explicitly encode known notions of geometry, instead of having the network implicitly learn them from data. However, performing eigendecomposition within a network requires the ability to differentiate this operation. Unfortunately, while theoretically doable, this introduces numerical instability in the optimization process in practice. In this paper, we introduce an eigendecomposition-free approach to training a deep network whose loss depends on the eigenvector corresponding to a zero eigenvalue of a matrix predicted by the network. We demonstrate on several tasks, including keypoint matching and 3D pose estimation, that our approach is much more robust than explicit differentiation of the eigendecomposition, It has better convergence properties and yields state-of-the-art results on both tasks.