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Stochastic Primal-Dual Three Operator Splitting Algorithm with Extension to Equivariant Regularization-by-Denoising

Junqi Tang, Matthias Ehrhardt, Carola-Bibiane Schönlieb

TL;DR

This work extends stochastic primal-dual methods to convex three-term composites by introducing TOS-SPDHG, a three-operator splitting of the saddle-point reformulation for problems of the form $f(Ax)+g(x)+h(x)$. It accommodates non-scalar preconditioners and arbitrary sampling, and integrates a smooth term with Lipschitz gradient, achieving an ergodic convergence rate of $O(1/K)$. The authors further adapt the framework to regularization-by-denoising (RED) priors, proposing two variants, TOS-SPDHG-RED and TOS-SPDHG-eRED, including an equivariant denoiser approach to enhance stability and performance. Numerical experiments in imaging inverse problems, particularly X-ray CT, demonstrate faster convergence than Condat-Vu and show that RED-based extensions improve reconstruction quality, especially in low-dose settings. The work provides a theoretically grounded, practically effective solver for large-scale three-term convex optimization with flexible priors in imaging applications.

Abstract

In this work we propose a stochastic primal-dual three-operator splitting algorithm (TOS-SPDHG) for solving a class of convex three-composite optimization problems. Our proposed scheme is a direct three-operator splitting extension of the SPDHG algorithm [Chambolle et al. 2018]. We provide theoretical convergence analysis showing ergodic $O(1/K)$ convergence rate, and demonstrate the effectiveness of our approach in imaging inverse problems. Moreover, we further propose TOS-SPDHG-RED and TOS-SPDHG-eRED which utilizes the regularization-by-denoising (RED) framework to leverage pretrained deep denoising networks as priors.

Stochastic Primal-Dual Three Operator Splitting Algorithm with Extension to Equivariant Regularization-by-Denoising

TL;DR

This work extends stochastic primal-dual methods to convex three-term composites by introducing TOS-SPDHG, a three-operator splitting of the saddle-point reformulation for problems of the form . It accommodates non-scalar preconditioners and arbitrary sampling, and integrates a smooth term with Lipschitz gradient, achieving an ergodic convergence rate of . The authors further adapt the framework to regularization-by-denoising (RED) priors, proposing two variants, TOS-SPDHG-RED and TOS-SPDHG-eRED, including an equivariant denoiser approach to enhance stability and performance. Numerical experiments in imaging inverse problems, particularly X-ray CT, demonstrate faster convergence than Condat-Vu and show that RED-based extensions improve reconstruction quality, especially in low-dose settings. The work provides a theoretically grounded, practically effective solver for large-scale three-term convex optimization with flexible priors in imaging applications.

Abstract

In this work we propose a stochastic primal-dual three-operator splitting algorithm (TOS-SPDHG) for solving a class of convex three-composite optimization problems. Our proposed scheme is a direct three-operator splitting extension of the SPDHG algorithm [Chambolle et al. 2018]. We provide theoretical convergence analysis showing ergodic convergence rate, and demonstrate the effectiveness of our approach in imaging inverse problems. Moreover, we further propose TOS-SPDHG-RED and TOS-SPDHG-eRED which utilizes the regularization-by-denoising (RED) framework to leverage pretrained deep denoising networks as priors.
Paper Structure (8 sections, 3 theorems, 38 equations, 3 figures)

This paper contains 8 sections, 3 theorems, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Stochastic primal-dual hybrid gradient and three operator splitting
  3. Extension to Equivariant RED Priors
  4. Convergence Analysis for TOS-SPDHG
  5. Convergence analysis
  6. Convergence proof
  7. Numerical Experiment
  8. Conclusion

Key Result

Lemma 2

Assuming that $h$ has $L$-Lipschitz continuous gradients, and both $f$, $g$, and $h$ are proper convex lower-semicontinous functions, for the deterministic updates we have:

Figures (3)

  • Figure 1: Sparse-view X-ray CT Reconstruction results for Condat-Vu and TOS-SPDHG, termintate at 150th epoch. For the first example we use least-squares as data-fit, where for the second example we use the KL divergence.
  • Figure 2: Low-dose sparse-view X-ray CT Reconstruction results for Condat-Vu, TOS-SPDHG-RED and TOS-SPDHG-eRED, termintate at 75th epoch. We choose DnCNN as the denoiser.
  • Figure 3: Low-dose sparse-view X-ray CT Reconstruction results for Condat-Vu, TOS-SPDHG-RED and TOS-SPDHG-eRED, termintate at 75th epoch. We choose DnCNN as the denoiser.

Theorems & Definitions (4)

  • Definition 1
  • Lemma 2
  • Lemma 3
  • Theorem 4