Differentiable Gaussianization Layers for Inverse Problems Regularized by Deep Generative Models

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 an inversion process, particularly in the presence of data noise and inaccurate forward models. In such cases, deep generative models are ineffective in attaining high-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 tested and validated 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, and achieved state-of-the-art performance in terms of accuracy and consistency.