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Stability Bounds for the Unfolded Forward-Backward Algorithm

Created by
  • Haebom

Author

Emilie Chouzenoux, Cecile Della Valle, Jean-Christophe Pesquet

Outline

We consider a neural network architecture designed to solve inverse problems with linear and known degradation operators. This architecture is constructed by unfolding a forward-backward algorithm derived by minimizing an objective function that combines a data fidelity term, a Tikhonov-type regularization term, and a potentially non-smooth convex penalty. We theoretically analyze the robustness of this inverse method to input perturbations. This robustness is guaranteed by principles of inverse problem theory, ensuring both continuity and resilience to small noise. This is an important property, considering that deep neural networks are vulnerable to adversarial perturbations. The main novelty of this study lies in investigating the robustness of the proposed network to biases in representing observed data in the inverse problem. We also provide numerical examples of the analytical Lipschitz bound derived from the analysis.

Takeaways, Limitations

Takeaways:
Designing robust neural network architectures for solving linear inverse problems.
Theoretical analysis of robustness to input perturbations and especially bias perturbations.
A design that adheres to the principles of inverse problem theory.
Numerical examples of analytical Lipschitz bounds are provided.
Limitations:
Limited to linear and known degradation operators.
Relying on Tikhonov regularization and potentially non-smooth convex penalties.
Generalization and performance evaluation for real-world complex problems are needed.
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