[Paper Review] Invertible generative models for inverse problems: mitigating representation error and dataset bias

Muhammad Asim, Max Daniels+3 more|arXiv (Cornell University)|May 28, 2019

Generative Adversarial Networks and Image Synthesis74 citations

TL;DR

The paper shows that invertible neural networks can serve as priors for imaging inverse problems, mitigating representation error and dataset bias, and outperforming GAN priors in several tasks while providing theoretical recovery bounds for linear invertible generators.

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.

Motivation & Objective

• Motivate the use of invertible generative models as zero-representation-error priors for imaging inverse problems.
• Evaluate denoising, compressive sensing, and inpainting performance of invertible priors against GANs and unlearned priors.
• Demonstrate bias mitigation and robustness to out-of-distribution images due to zero representation error.
• Provide a theoretical bound on recovery error for linear invertible generators in compressive sensing.

Proposed method

• Use a pretrained Glow-based invertible generator G: R^n -> R^n with x = G(z) and z = G^{-1}(x).
• Formulate inverse problems as latent-space optimizations: (1) denoising with regularization on latent code ||z||^2 and data-fidelity, (2) compressive sensing with data-fidelity only (gamma = 0) initialized at z0 = 0.
• Solve (2) via L-BFGS for Glow and use Adam for Deep Decoder, DCGAN, and PGGAN; compare against BM3D and sparsity-based priors.
• Train Glow on CelebA; fix prior for all inverse problems; compare with in-distribution and out-of-distribution datasets (CelebA-HQ vs FFHQ).
• Provide a theoretical analysis for the linear invertible generator case: bounds on expected recovery error in terms of singular values of G.

Experimental results

Research questions

• RQ1 Can invertible priors achieve higher quality reconstructions than GAN priors and unlearned priors across denoising, compressive sensing, and inpainting?
• RQ2 Do invertible priors mitigate dataset bias and remain effective on out-of-distribution images due to zero representation error?
• RQ3 What are the theoretical recovery guarantees for linear invertible generative models under compressive sensing?

Key findings

• Invertible priors can yield sharper denoising results with higher PSNR than BM3D when regularized appropriately.
• For compressive sensing on in-distribution images, Glow priors outperform GAN priors (DCGAN/PGGAN) and Deep Decoder across a wide range of undersampling ratios.
• Glow priors show graceful performance decay on out-of-distribution images and can surpass GANs with low latent dimensionality, sometimes outperforming Deep Decoder when measurements are sufficient.
• A theoretical bound is established: for a linear invertible generator, the expected squared recovery error with m measurements is between sum of squares of non-top singular values and m times the sum of squares of the next set of singular values.
• Compared to dataset-biased GANs (e.g., PGGAN trained on CelebA), Glow mitigates biases and can recover features underrepresented in the training set.