mlsanityData to 7 Oct 2026

Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation

arXiv 2402.01052 ↗First posted 1 Feb 2024math.OC cs.CV cs.LGcited by 12

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Abstract (arXiv)

Variational regularisation is the primary method for solving inverse problems, and recently there has been considerable work leveraging deeply learned regularisation for enhanced performance. However, few results exist addressing the convergence of such regularisation, particularly within the context of critical points as opposed to global minimisers. In this paper, we present a generalised formulation of convergent regularisation in terms of critical points, and show that this is achieved by a class of weakly convex regularisers. We prove convergence of the primal-dual hybrid gradient method for the associated variational problem, and, given a Kurdyka-Lojasiewicz condition, an O(logk/k) ergodic convergence rate. Finally, applying this theory to learned regularisation, we prove universal approximation for input weakly convex neural networks (IWCNN), and show empirically that IWCNNs can lead to improved performance of learned adversarial regularisers for computed tomography (CT) reconstruction.

Key facts. Cited by 12 papers in the map (data through 7 Oct 2026). Builds on 40 of its 85 references that are in the map. First posted on arXiv 1 Feb 2024.

Builds on

40 of 85 references.

References · 40

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28 more
  1. 2022Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularizationcited by 79
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Built on by

Cited by year

Citing papers per year: 2024: 4; 2025: 8.

12 papers.

  1. 2024Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groupscited by 23
  2. 2025Learning Regularization Functionals for Inverse Problems: A Comparative Studycited by 12
  3. 2024Benchmarking learned algorithms for computed tomography image reconstruction taskscited by 6
  4. 2024Stability of Data-Dependent Ridge-Regularization for Inverse Problemscited by 5
  5. 2024The Star Geometry of Critic-Based Regularizer Learningcited by 3
  6. 2025A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithmcited by 3
  7. 2025Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoiserscited by 2
  8. 2025On Information Geometry and Iterative Optimization in Model Compression: Operator Factorizationcited by 1
  9. 2025WEEP: A Differentiable Nonconvex Sparse Regularizer via Weakly-Convex Envelopecited by 1
  10. 2025RINS-T: Robust Implicit Neural Solvers for Time Series Linear Inverse Problemscited by 1
  11. 2025MAP Image Recovery with Guarantees using Locally Convex Multi-Scale Energy (LC-MUSE) Modelcited by 0
  12. 2025Optimal Regularization Under Uncertainty: Distributional Robustness and Convexity Constraintscited by 0