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Stochastic Orthogonal Regularization for deep projective priors

Ali Joundi, Yann Traonmilin, Alasdair Newson

TL;DR

This work tackles instability in deep projective priors used within generalized projected gradient descent (GPGD) for imaging inverse problems. It introduces Stochastic Orthogonal Regularization (SOR) to push projections toward near-orthogonality, enabling linear convergence guarantees under a restricted isometry property. The authors formalize bounds on the restricted Lipschitz constant via orthogonality metrics and validate the approach with autoencoders on MNIST and DRUNET denoisers on CelebA, demonstrating faster convergence with minimal PSNR/SSIM trade-offs. Overall, SOR improves robustness and speed of GPGD across diverse inverse problems and priors, while highlighting theoretical and practical limitations and future directions.

Abstract

Many crucial tasks of image processing and computer vision are formulated as inverse problems. Thus, it is of great importance to design fast and robust algorithms to solve these problems. In this paper, we focus on generalized projected gradient descent (GPGD) algorithms where generalized projections are realized with learned neural networks and provide state-of-the-art results for imaging inverse problems. Indeed, neural networks allow for projections onto unknown low-dimensional sets that model complex data, such as images. We call these projections deep projective priors. In generic settings, when the orthogonal projection onto a lowdimensional model set is used, it has been shown, under a restricted isometry assumption, that the corresponding orthogonal PGD converges with a linear rate, yielding near-optimal convergence (within the class of GPGD methods) in the classical case of sparse recovery. However, for deep projective priors trained with classical mean squared error losses, there is little guarantee that the hypotheses for linear convergence are satisfied. In this paper, we propose a stochastic orthogonal regularization of the training loss for deep projective priors. This regularization is motivated by our theoretical results: a sufficiently good approximation of the orthogonal projection guarantees linear stable recovery with performance close to orthogonal PGD. We show experimentally, using two different deep projective priors (based on autoencoders and on denoising networks), that our stochastic orthogonal regularization yields projections that improve convergence speed and robustness of GPGD in challenging inverse problem settings, in accordance with our theoretical findings.

Stochastic Orthogonal Regularization for deep projective priors

TL;DR

This work tackles instability in deep projective priors used within generalized projected gradient descent (GPGD) for imaging inverse problems. It introduces Stochastic Orthogonal Regularization (SOR) to push projections toward near-orthogonality, enabling linear convergence guarantees under a restricted isometry property. The authors formalize bounds on the restricted Lipschitz constant via orthogonality metrics and validate the approach with autoencoders on MNIST and DRUNET denoisers on CelebA, demonstrating faster convergence with minimal PSNR/SSIM trade-offs. Overall, SOR improves robustness and speed of GPGD across diverse inverse problems and priors, while highlighting theoretical and practical limitations and future directions.

Abstract

Many crucial tasks of image processing and computer vision are formulated as inverse problems. Thus, it is of great importance to design fast and robust algorithms to solve these problems. In this paper, we focus on generalized projected gradient descent (GPGD) algorithms where generalized projections are realized with learned neural networks and provide state-of-the-art results for imaging inverse problems. Indeed, neural networks allow for projections onto unknown low-dimensional sets that model complex data, such as images. We call these projections deep projective priors. In generic settings, when the orthogonal projection onto a lowdimensional model set is used, it has been shown, under a restricted isometry assumption, that the corresponding orthogonal PGD converges with a linear rate, yielding near-optimal convergence (within the class of GPGD methods) in the classical case of sparse recovery. However, for deep projective priors trained with classical mean squared error losses, there is little guarantee that the hypotheses for linear convergence are satisfied. In this paper, we propose a stochastic orthogonal regularization of the training loss for deep projective priors. This regularization is motivated by our theoretical results: a sufficiently good approximation of the orthogonal projection guarantees linear stable recovery with performance close to orthogonal PGD. We show experimentally, using two different deep projective priors (based on autoencoders and on denoising networks), that our stochastic orthogonal regularization yields projections that improve convergence speed and robustness of GPGD in challenging inverse problem settings, in accordance with our theoretical findings.
Paper Structure (27 sections, 3 theorems, 31 equations, 20 figures, 7 tables)

This paper contains 27 sections, 3 theorems, 31 equations, 20 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work:
  4. Stochastic orthogonal regularization and linear convergence of generalized projected gradient descent
  5. Definitions
  6. Restricted Lipschitz constants of approximate orthogonal projections
  7. Stochastic orthogonal regularization of deep projective priors
  8. Experimental results
  9. SOR of autoencoders DPP on MNIST
  10. SOR of Plug and Play DPP on CelebA images
  11. Discussions and Limitations
  12. Conclusion
  13. Proof of Theorem \ref{['th:gen_convevergence']}
  14. Proof of Theorem \ref{['theorem:general_rlc']}
  15. Proof of Theorem \ref{['theorem:eta_L']}
  16. ...and 12 more sections

Key Result

Theorem 2.5

Consider the observation model (eq:eq_pb_inverse). Let $\Sigma \subset \mathbb{R}^n$. Suppose $\gamma \mathbf{A}^T\mathbf{A}$ has a RIC $\delta:=\delta_\Sigma(\gamma \mathbf{A}^T\mathbf{A})$. Suppose $P_\Sigma$ is a generalized projection with the restricted $\beta$-Lipschitz property with respect t

Figures (20)

  • Figure 1: Recovery of MNIST images from a super-resolution inverse problem with different PGDs using autoencoders weighted differently. The more lambda increases, the quicker the convergence, and this is without a significant loss of recovery PSNR.
  • Figure 1: PSNR of MNIST recovery and $\psi_P$ for four different datasets. $\psi_P$ is decreasing overall for the different datasets with respect to $\lambda$ while the PSNR is only slightly degraded.
  • Figure 1: Graph and visual results of the evolution of the recovered MRI images with the increase of the inpainting ratio for the different denoisers. In higher ratios, the regularized denoiser performs better than the non-regularized denoiser.
  • Figure 1: Recovery of MNIST images from a noiseless super resolution inverse problem with different PGDs using autoencoders weighted differently
  • Figure 1: Inpainting of an image with an inpainting ratio of 0.7 and noise level of 0. Visually, SOR produces more textures compared to RED which tends to average the image. However, it adds more artifacts.
  • ...and 15 more figures

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2: Generalized projection
  • Definition 2.3: Orthogonal projection
  • Definition 2.4: Restricted Lipschitz property
  • Theorem 2.5: Stable linear recovery of low-dimensional models
  • Theorem 2.6
  • Theorem 2.7
  • Proof 1
  • Proof 2: Proof of theorem \ref{['theorem:general_rlc']}
  • Proof 3: Proof of theorem \ref{['theorem:eta_L']}