Invertible generative models for inverse problems: mitigating representation error and dataset bias

TL;DR

The paper argues that invertible neural networks, exemplified by Glow, provide zero representation error as priors for imaging inverse problems, enabling accurate recovery across denoising, compressive sensing, and inpainting, including out-of-distribution data. It formalizes a latent-space empirical risk approach, showing that optimizing in latent space with a data-fit term (and selective regularization) can outperform traditional and GAN-based priors, even when measurements are scarce. A key theoretical contribution gives bounds on the expected recovery error for linear invertible generators in terms of the singular values of the generator, illustrating how rapid decay can drive error to zero as measurements approach the signal dimension. Empirically, Glow demonstrates superior PSNR/SSIM performance and qualitative reconstructions compared with GAN priors and unlearned priors, while mitigating dataset bias and maintaining robustness to distribution shift. The work suggests a practical, generalizable paradigm for inverse problems with strong implications for medical and scientific imaging, while highlighting avenues for architecture improvements to balance invertibility, training cost, and inversion speed.

Abstract

Trained generative models have shown remarkable performance as priors for inverse problems in imaging -- for example, Generative Adversarial Network priors permit recovery of test images from 5-10x fewer measurements than sparsity priors. Unfortunately, these models may be unable to represent any particular image because of architectural choices, mode collapse, and bias in the training dataset. In this paper, we demonstrate that invertible neural networks, which have zero representation error by design, can be effective natural signal priors at inverse problems such as denoising, compressive sensing, and inpainting. Given a trained generative model, we study the empirical risk formulation of the desired inverse problem under a regularization that promotes high likelihood images, either directly by penalization or algorithmically by initialization. For compressive sensing, invertible priors can yield higher accuracy than sparsity priors across almost all undersampling ratios, and due to their lack of representation error, invertible priors can yield better reconstructions than GAN priors for images that have rare features of variation within the biased training set, including out-of-distribution natural images. We additionally compare performance for compressive sensing to unlearned methods, such as the deep decoder, and we establish theoretical bounds on expected recovery error in the case of a linear invertible model.

Figures (29)

• Figure 1: We train an invertible generative model with CelebA images (including those shown). When used as a prior for compressive sensing, it can yield higher quality image reconstructions than Lasso and a trained DCGAN, even on out-of-distribution images. Note that the DCGAN reflects biases of the training set by removing the man's glasses and beard, whereas our invertible prior does not.
• Figure 2: An Affine Coupling layer applies an affine transformation to half of the input data, here $x_1$. The parameters of the affine transformations, $s$ and $t$, can depend in a complex, learned way on the other half of the input data. The model can be inverted, even though $s$ and $t$ themselves are not invertible.
• Figure 3: Recovered PSNR values as a function of $\gamma$ for denoising by the Glow and DCGAN priors. Denoising results are averaged over $N=50$ in-distribution test set images. For reference, we show the average PSNRs of the original noisy images, after applying BM3D, and under the Glow prior in the noiseless case ($\sigma = 0$).
• Figure 4: Denoising results using the Glow prior, the DCGAN prior, and BM3D at noise level $\sigma = 0.1$. Note that the Glow prior gives a sharper image than BM3D in these cases.
• Figure 5:

Theorems & Definitions (6)

• Theorem 1
• proof : Proof of Theorem \ref{['main_thm']}
• Theorem 2: Minor variant of Theorem 10.5 in Troppetal2011
• Lemma 1
• proof
• Lemma 2: Proposition 8.4 in Troppetal2011