# Differentiable SVD based on Moore-Penrose Pseudoinverse for Inverse Imaging Problems

arXiv: https://arxiv.org/abs/2411.14141 · HTML: https://mlsanity.com/paper/2411.14141/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 45 references that are in the map.
- First posted on arXiv 21 Nov 2024.

## Builds on (most cited first, 3 of 3)

- [Torch.manual_seed(3407) is all you need: On the influence of random seeds in deep learning architectures for computer vision](https://mlsanity.com/paper/2109.08203/) (2021)
- [OCMR (v1.0)--Open-Access Multi-Coil k-Space Dataset for Cardiovascular Magnetic Resonance Imaging](https://mlsanity.com/paper/2008.03410/) (2020)
- [Deep MRI Reconstruction: Unrolled Optimization Algorithms Meet Neural Networks](https://mlsanity.com/paper/1907.11711/) (2019)

## Built on by (most cited first, 3 of 3)

- [USCTNet: A deep unfolding nuclear-norm optimization solver for physically consistent HSI reconstruction](https://mlsanity.com/paper/2509.10651/) (2025)
- [Computation-Aware Kalman Filtering with Model Selection for Neural Dynamics](https://mlsanity.com/paper/2606.01468/) (2026)
- [Fast Differentiable SVD on GPU via Polar Decomposition](https://mlsanity.com/paper/2609.32505/) (2026)

## Abstract (arXiv)

Low-rank regularization-based deep unrolling networks have achieved remarkable success in various inverse imaging problems (IIPs). However, the singular value decomposition (SVD) is non-differentiable when duplicated singular values occur, leading to severe numerical instability during training. In this paper, we propose a differentiable SVD based on the Moore-Penrose pseudoinverse to address this issue. To the best of our knowledge, this is the first work to provide a comprehensive analysis of the differentiability of the trivial SVD. Specifically, we show that the non-differentiability of SVD is essentially due to an underdetermined system of linear equations arising in the derivation process. We utilize the Moore-Penrose pseudoinverse to solve the system, thereby proposing a differentiable SVD. A numerical stability analysis in the context of IIPs is provided. Experimental results in color image compressed sensing and dynamic MRI reconstruction show that our proposed differentiable SVD can effectively address the numerical instability issue while ensuring computational precision. Code is available at https://github.com/yhao-z/SVD-inv.