# Quasi-SVD: Learning a Lie-constrained matrix factorisation for real-time imaging

arXiv: https://arxiv.org/abs/2607.25967 · HTML: https://mlsanity.com/paper/2607.25967/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 52 references that are in the map.
- First posted on arXiv 28 Jul 2026.

## Builds on (most cited first, 21 of 21)

- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Gaussian Splatting SLAM](https://mlsanity.com/paper/2312.06741/) (2023)
- [SVD-LLM: Truncation-aware Singular Value Decomposition for Large Language Model Compression](https://mlsanity.com/paper/2403.07378/) (2024)
- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [Efficient Riemannian Optimization on the Stiefel Manifold via the Cayley Transform](https://mlsanity.com/paper/2002.01113/) (2020)
- [Deep Spectral Clustering using Dual Autoencoder Network](https://mlsanity.com/paper/1904.13113/) (2019)
- [Randomized Block Krylov Methods for Stronger and Faster Approximate Singular Value Decomposition](https://mlsanity.com/paper/1504.05477/) (2015)
- [Tensor SVD: Statistical and Computational Limits](https://mlsanity.com/paper/1703.02724/) (2017)
- [Learning How To Robustly Estimate Camera Pose in Endoscopic Videos](https://mlsanity.com/paper/2304.08023/) (2023)
- [Tangent Space Backpropagation for 3D Transformation Groups](https://mlsanity.com/paper/2103.12032/) (2021)
- [Deep K-SVD Denoising](https://mlsanity.com/paper/1909.13164/) (2019)
- [QSVD: Efficient Low-rank Approximation for Unified Query-Key-Value Weight Compression in Low-Precision Vision-Language Models](https://mlsanity.com/paper/2510.16292/) (2025)
- [Operator SVD with Neural Networks via Nested Low-Rank Approximation](https://mlsanity.com/paper/2402.03655/) (2024)
- [DiTASK: Multi-Task Fine-Tuning with Diffeomorphic Transformations](https://mlsanity.com/paper/2502.06029/) (2025)
- [Learned SVD: solving inverse problems via hybrid autoencoding](https://mlsanity.com/paper/1912.10840/) (2019)
- [Efficient GPU implementation of randomized SVD and its applications](https://mlsanity.com/paper/2110.03423/) (2021)
- [Background Subtraction via Generalized Fused Lasso Foreground Modeling](https://mlsanity.com/paper/1504.03707/) (2015)
- [Physically Consistent Image Augmentation for Deep Learning in Mueller Matrix Polarimetry](https://mlsanity.com/paper/2411.07918/) (2024)
- [A hierarchically blocked Jacobi SVD algorithm for single and multiple graphics processing units](https://mlsanity.com/paper/1401.2720/) (2014)
- [RF-ULM: Ultrasound Localization Microscopy Learned from Radio-Frequency Wavefronts](https://mlsanity.com/paper/2310.01545/) (2023)
- [Near-Real-Time Mueller Polarimetric Image Processing for Neurosurgical Intervention](https://mlsanity.com/paper/2403.00893/) (2024)

## Abstract (arXiv)

Singular Value Decomposition (SVD) underlies matrix factorisation tasks across many fields, with imaging applications demanding real-time processing. Yet SVD algorithms are inherently sequential, constraining real-time GPU throughput and limit online deployment in imaging pipelines. This study introduces a fully parallelized matrix factorization framework for GPUs by enforcing matrix orthogonality on left singular vectors via Lie-parametrised algebra and recovering the remaining components through soft constraints. This asymmetric constraint design enables an efficient parallel and provably valid decomposition, achieves high reconstruction fidelity and substantially accelerates computation relative to the exact SVD, with real-time throughput exceeding standard video frame rates. Performance is evaluated on multiple imaging tasks spanning complementary computational regimes: (1) spatio-temporal background subtraction for ultrasound localisation microscopy, requiring high-dimensional matrix separation, (2) Mueller matrix polarimetry for neurosurgical tissue characterisation, requiring massive batch processing of small matrices, and (3) an MNIST denoising benchmark at an intermediate scale with known ground truth. Across regimes and instruments, the proposed framework demonstrates robust domain transfer at various matrix scales, sufficient for live image-guided workflows that classical solvers cannot currently support in these settings. By prioritising downstream reconstruction fidelity over exact spectral recovery, the proposed SVD framework makes structured matrix factorisation practical for real-time processing.