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To be or not to be stable, that is the question: understanding neural networks for inverse problems

Davide Evangelista, James Nagy, Elena Morotti, Elena Loli Piccolomini

TL;DR

The work analyzes stability-accuracy trade-offs in neural-network Solvers for linear inverse problems with Gaussian noise, establishing a formal framework of reconstructors, $\eta^{-1}$-accuracy, and $\epsilon$-stability. It introduces ReNN, a ground-truth-free training paradigm, and stabilizer-based approaches (StNN/StReNN) that integrate model-based pre-processing to mitigate noise sensitivity. Theoretical results link stability to local Lipschitz properties and iterative stabilizers (e.g., Tikhonov/CGLS sketches), while experiments on image deblurring demonstrate substantial stability improvements with controlled accuracy loss, and show ReNN's robustness when ground-truth data are unavailable. Collectively, these methods offer principled, practically effective tools for stable, high-quality image reconstruction in noisy linear inverse problems, with potential applicability to medical imaging and other ill-posed tasks.

Abstract

The solution of linear inverse problems arising, for example, in signal and image processing is a challenging problem since the ill-conditioning amplifies, in the solution, the noise present in the data. Recently introduced algorithms based on deep learning overwhelm the more traditional model-based approaches in performance, but they typically suffer from instability with respect to data perturbation. In this paper, we theoretically analyze the trade-off between stability and accuracy of neural networks, when used to solve linear imaging inverse problems for not under-determined cases. Moreover, we propose different supervised and unsupervised solutions to increase the network stability and maintain a good accuracy, by means of regularization properties inherited from a model-based iterative scheme during the network training and pre-processing stabilizing operator in the neural networks. Extensive numerical experiments on image deblurring confirm the theoretical results and the effectiveness of the proposed deep learning-based approaches to handle noise on the data.

To be or not to be stable, that is the question: understanding neural networks for inverse problems

TL;DR

The work analyzes stability-accuracy trade-offs in neural-network Solvers for linear inverse problems with Gaussian noise, establishing a formal framework of reconstructors, -accuracy, and -stability. It introduces ReNN, a ground-truth-free training paradigm, and stabilizer-based approaches (StNN/StReNN) that integrate model-based pre-processing to mitigate noise sensitivity. Theoretical results link stability to local Lipschitz properties and iterative stabilizers (e.g., Tikhonov/CGLS sketches), while experiments on image deblurring demonstrate substantial stability improvements with controlled accuracy loss, and show ReNN's robustness when ground-truth data are unavailable. Collectively, these methods offer principled, practically effective tools for stable, high-quality image reconstruction in noisy linear inverse problems, with potential applicability to medical imaging and other ill-posed tasks.

Abstract

The solution of linear inverse problems arising, for example, in signal and image processing is a challenging problem since the ill-conditioning amplifies, in the solution, the noise present in the data. Recently introduced algorithms based on deep learning overwhelm the more traditional model-based approaches in performance, but they typically suffer from instability with respect to data perturbation. In this paper, we theoretically analyze the trade-off between stability and accuracy of neural networks, when used to solve linear imaging inverse problems for not under-determined cases. Moreover, we propose different supervised and unsupervised solutions to increase the network stability and maintain a good accuracy, by means of regularization properties inherited from a model-based iterative scheme during the network training and pre-processing stabilizing operator in the neural networks. Extensive numerical experiments on image deblurring confirm the theoretical results and the effectiveness of the proposed deep learning-based approaches to handle noise on the data.
Paper Structure (22 sections, 16 theorems, 86 equations, 5 figures, 4 tables, 1 algorithm)

This paper contains 22 sections, 16 theorems, 86 equations, 5 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Structure of the paper
  4. Reconstructors for the solution of linear inverse problems
  5. Accuracy vs. stability trade-off
  6. A sufficient condition for stability
  7. Stabilizers in the solution of linear inverse problems
  8. Stabilizers and properties
  9. Tikhonov stabilizers
  10. Neural networks for the solution of linear inverse problems
  11. Parameter-dependent families of reconstructors
  12. Neural Networks as reconstructors: the NN approach
  13. Regularized NN-based reconstructors: the ReNN approach
  14. Stabilization on NN and ReNN: St- approaches
  15. Experimental setup
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $\Psi: \mathbb{R}^m \to \mathbb{R}^n$ be an $\eta^{-1}$-accurate reconstructor. Then, for any $\boldsymbol{x}^{gt} \in \mathcal{X}$ and for any $\epsilon > 0$, $\exists \: \tilde{\boldsymbol{e}} \in \mathbb{R}^m$ with $|| \tilde{\boldsymbol{e}} || \leq \epsilon$ such that:

Figures (5)

  • Figure 1: Graphical representation of the $\epsilon$-stability and $\epsilon$-instability for an $\eta^{-1}$-accurate reconstructor.
  • Figure 2: A schematic representation of the proposed methods.
  • Figure 3: Results obtained by the NN and StNN reconstructors on a single test image $y^{\delta}$ with $delta=0$ (first row) and $\delta=0.01$ (second row). The ground truth clean image is also reported for reference.
  • Figure 5: Blurred noisy input image $\boldsymbol{y}^{\delta}$ ($\delta = 0.06$) on the top left and examples of reconstruction obtained by the iNN, StiNN, ReNN, StReNN and Tikhonov methods on a test image.
  • Figure 6: (a) Plots of the empirical error yielded by iNN, StiNN, ReNN, StReNN and Tikhonov reconstructors for increasing values of $\delta$ in the test images. (b) Boxplots over the $T=20$ executions.

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2
  • Example 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.2
  • Example 2.3
  • Lemma 2.1
  • proof
  • ...and 33 more