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.

翻译：基于低秩正则化的深度展开网络在各种逆成像问题中取得了显著成功。然而，当出现重复奇异值时，奇异值分解具有不可微性，导致训练过程中出现严重的数值不稳定问题。本文提出一种基于Moore-Penrose伪逆的可微SVD方法以解决该问题。据我们所知，这是首个对平凡SVD可微性进行全面分析的工作。具体而言，我们证明SVD的不可微性本质上源于推导过程中出现的欠定线性方程组。我们利用Moore-Penrose伪逆求解该方程组，从而提出可微SVD方法。本文提供了在逆成像问题背景下的数值稳定性分析。彩色图像压缩感知和动态MRI重建的实验结果表明，所提出的可微SVD方法在保证计算精度的同时，能有效解决数值不稳定问题。代码发布于https://github.com/yhao-z/SVD-inv。