QUASI-SVD: LEARNING A LIE-CONSTRAINED MATRIX FACTORISATION FOR REAL-TIME IMAGING

Classical Singular Value Decomposition (SVD) is a fundamental bottleneck in real-time medical imaging, such as ultrasound localisation microscopy and Mueller matrix polarimetry. Existing SVD algorithms rely on sequential operations like Householder reflections, which prevent efficient GPU parallelization and limit throughput in clinical pipelines. This paper asks whether a differentiable, Lie-constrained matrix factorization can provide the necessary speed for real-time deployment without sacrificing the reconstruction quality required for clinical decision-making.

The author introduces Quasi-SVD, a framework that replaces the traditional, sequential SVD process with an asymmetric, parallel-friendly design. Instead of enforcing orthogonality on both singular-vector factors, Quasi-SVD uses the matrix exponential over a skew-symmetric Lie algebra to guarantee exact orthogonality for one factor. The remaining components are recovered through soft constraints. This approach allows the decomposition to be computed in parallel on GPUs, bypassing the iterative, sequential dependencies of classical solvers. The framework is evaluated across two distinct computational regimes: high-dimensional matrix separation for ultrasound and massive batch processing of small matrices for polarimetry.

Quasi-SVD achieves a 3–20x speedup compared to standard solvers like cuSOLVER and randomized SVD, consistently exceeding 25 frames per second (FPS) at clinical matrix scales. Despite the approximation, the framework maintains high reconstruction fidelity, with structural similarity index (SSIM) scores ranging from 0.89 to 0.94. By leveraging Lie-theoretic constraints, the method avoids the numerical drift typically associated with purely learning-based SVD approximations, making it a robust alternative for real-time, image-guided workflows.