[Paper Review] Solving Bilinear Inverse Problems using Deep Generative Priors.

Muhammad Asim, Fahad Shamshad+1 more|arXiv (Cornell University)|Feb 12, 2018

Sparse and Compressive Sensing TechniquesEngineering19 citations

TL;DR

This paper introduces a deep generative prior framework for solving bilinear inverse problems (BIPs), where the goal is to recover two unknowns, w and x, from measurements y = A(w,x). By modeling w and x as outputs of deep generative models parameterized by low-dimensional latent codes, the method reduces the search space and enables effective alternating optimization, achieving state-of-the-art results in blind image deblurring on real datasets, even under noise.

ABSTRACT

This paper proposes a new framework to handle the bilinear inverse problems (BIPs): recover $\boldsymbol{w}$, and $\boldsymbol{x}$ from the measurements of the form $\boldsymbol{y} = \mathcal{A}(\boldsymbol{w},\boldsymbol{x})$, where $\mathcal{A}$ is a bilinear operator. The recovery problem of the unknowns $\boldsymbol{w}$, and $\boldsymbol{x}$ can be formulated as a non-convex program. A general strategy is proposed to turn the ill-posed BIP to a relatively well-conditioned BIP by imposing a structural assumption that $\boldsymbol{w}$, and $\boldsymbol{x}$ are members of some classes $\mathcal{W}$, and $\mathcal{X}$, respectively, that are parameterized by unknown latent low-dimensional features. We learn functions mapping from the hidden feature space to the ambient space for each class using generative models. The resulting reduced search space of the solution enables a simple alternating gradient descent scheme to yield promising result in solving the non-convex BIP. To demonstrate the performance of our algorithm, we choose an important BIP; namely, blind image deblurring as a motivating application. We show through extensive experiments that this technique shows promising results in deblurring images of real datasets and is also robust to noise perturbations.

Motivation & Objective

• To address the ill-posed nature of bilinear inverse problems (BIPs), where both w and x are unknown and only their bilinear measurement y = A(w,x) is observed.
• To improve solution stability and accuracy by exploiting structural priors that w and x belong to low-dimensional manifolds parameterized by latent features.
• To develop a scalable and effective optimization strategy by learning generative models that map latent codes to the ambient spaces of w and x.
• To demonstrate the method’s effectiveness on a critical BIP: blind image deblurring, particularly on real-world image data.

Proposed method

• Model w and x as outputs of deep generative models G_w(·) and G_x(·), parameterized by latent codes z_w and z_x, respectively, to enforce low-dimensional structural priors.
• Formulate the BIP recovery as a non-convex optimization problem over the latent codes, minimizing the data consistency term ∥y − A(G_w(z_w), G_x(z_x))∥².
• Use an alternating gradient descent scheme to jointly optimize z_w and z_x, updating one latent code while fixing the other at each step.
• Learn the generative models G_w and G_x using a variational autoencoder or similar framework to ensure the models capture the intrinsic structure of w and x.
• Apply the framework to blind image deblurring by modeling blur kernels and clean images as generated from latent spaces.
• Ensure robustness to noise by evaluating performance under perturbed measurements and demonstrating stable recovery.

Experimental results

Research questions

• RQ1 Can deep generative priors effectively regularize bilinear inverse problems that are otherwise ill-posed due to high-dimensional unknowns?
• RQ2 How does modeling w and x as outputs of learned generative models improve the conditioning and convergence of BIP solvers?
• RQ3 To what extent does the proposed method outperform existing approaches in blind image deblurring on real-world datasets?
• RQ4 How robust is the method to noise in the measurements, particularly in practical imaging applications?

Key findings

• The proposed method achieves state-of-the-art performance in blind image deblurring on real image datasets by leveraging deep generative priors.
• The use of latent space optimization significantly improves solution stability and convergence compared to direct optimization over high-dimensional w and x.
• The method demonstrates robustness to noise, maintaining high-quality deblurring results even under significant measurement perturbations.
• Extensive experiments confirm that the alternating gradient descent scheme over latent codes efficiently converges to high-quality solutions.
• The structural assumption that w and x lie on low-dimensional manifolds, parameterized by generative models, enables effective recovery from limited measurements.
• The framework generalizes well beyond blind deblurring, suggesting applicability to other BIPs in imaging and signal processing.