Deep generative models such as GANs, normalizing flows, and diffusion models are powerful regularizers for inverse problems. They exhibit great potential for helping reduce ill-posedness and attain high-quality results. However, the latent tensors of such deep generative models can fall out of the desired high-dimensional standard Gaussian distribution during inversion, particularly in the presence of data noise and inaccurate forward models, leading to low-fidelity solutions. To address this issue, we propose to reparameterize and Gaussianize the latent tensors using novel differentiable data-dependent layers wherein custom operators are defined by solving optimization problems. These proposed layers constrain inverse problems to obtain high-fidelity in-distribution solutions. We validate our technique on three inversion tasks: compressive-sensing MRI, image deblurring, and eikonal tomography (a nonlinear PDE-constrained inverse problem) using two representative deep generative models: StyleGAN2 and Glow. Our approach achieves state-of-the-art performance in terms of accuracy and consistency.

翻译：深度生成模型（如生成对抗网络、归一化流与扩散模型）是逆问题的强效正则化工具，在缓解病态性并获取高质量结果方面展现出巨大潜力。然而，在逆问题求解过程中——尤其当存在数据噪声与不精确前向模型时——此类深度生成模型的潜在张量可能偏离理想的高维标准高斯分布，导致解的低保真度。针对此问题，我们提出通过新型可微分数据依赖层对潜在张量进行重参数化与高斯化，其中自定义算子通过求解优化问题来定义。所提出的层结构能够约束逆问题，从而获得高保真度的分布内解。我们使用两种代表性深度生成模型（StyleGAN2与Glow），在压缩感知MRI、图像去模糊及非线性PDE约束逆问题（走时层析成像）三种逆任务上验证了该方法。实验表明，本方法在精度与一致性方面均达到了当前最优性能。