  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.
