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Group Symmetry Enables Faster Optimization in Inverse Problems

Junqi Tang, Guixian Xu

TL;DR

The paper addresses fast optimization for linear inverse problems by exploiting group-symmetry structure. It introduces Group-PGD, a symmetry-aware gradient method that perturbs iterates with group actions and uses an augmented forward operator $A_{G_\star}$ to achieve improved conditioning. The main theoretical contribution is a convergence bound showing linear convergence of Group-PGD under a relaxed restricted-strong-convexity condition on $A_{G_\star}$, with an explicit rate $\alpha_{G^\star}=\kappa_c(1-\mu_{G^\star}/L)^{1/2}$ and an error term tied to symmetry perturbations $\varepsilon_{G^\star}$ and noise $w$. Numerical results on sparse-view CT confirm substantial acceleration over standard PGD, validating the approach and highlighting practical guidelines for choosing the symmetric subset $G_\star$.

Abstract

We prove for the first time that, if a linear inverse problem exhibits a group symmetry structure, gradient-based optimizers can be designed to exploit this structure for faster convergence rates. This theoretical finding demonstrates the existence of a special class of structure-adaptive optimization algorithms which are tailored for symmetry-structured inverse problems such as CT/MRI/PET, compressed sensing, and image processing applications such as inpainting/deconvolution, etc.

Group Symmetry Enables Faster Optimization in Inverse Problems

TL;DR

The paper addresses fast optimization for linear inverse problems by exploiting group-symmetry structure. It introduces Group-PGD, a symmetry-aware gradient method that perturbs iterates with group actions and uses an augmented forward operator to achieve improved conditioning. The main theoretical contribution is a convergence bound showing linear convergence of Group-PGD under a relaxed restricted-strong-convexity condition on , with an explicit rate and an error term tied to symmetry perturbations and noise . Numerical results on sparse-view CT confirm substantial acceleration over standard PGD, validating the approach and highlighting practical guidelines for choosing the symmetric subset .

Abstract

We prove for the first time that, if a linear inverse problem exhibits a group symmetry structure, gradient-based optimizers can be designed to exploit this structure for faster convergence rates. This theoretical finding demonstrates the existence of a special class of structure-adaptive optimization algorithms which are tailored for symmetry-structured inverse problems such as CT/MRI/PET, compressed sensing, and image processing applications such as inpainting/deconvolution, etc.
Paper Structure (6 sections, 2 theorems, 33 equations, 2 figures)

This paper contains 6 sections, 2 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hunting in the shadow: exploiting the group symmetry structure for acceleration
  3. Main result
  4. The proof of Theorem 3.1
  5. Numerical verification
  6. Conclusion

Key Result

Lemma 2.1

(Projection identities, oymak2017sharp) Given a set $\mathcal{K}$, $\mathcal{B}^d$ the unit ball in $\mathbb{R}^d$ and an orthogonal projection operator $\mathcal{P}_{\mathcal{K}}$, $x^\dagger \in \mathcal{K}$ and we have: and and where $\kappa_c = 1$ if $\mathcal{K}$ is convex, $\kappa_c = 2$ if $\mathcal{K}$ is nonconvex.

Figures (2)

  • Figure 1: Sparse-view fan-beam CT example, comparing PGD with Group-PGD
  • Figure 2: Extreme sparse-view fan-beam CT example, comparing PGD with Group-PGD

Theorems & Definitions (4)

  • Lemma 2.1
  • Theorem 3.1
  • Remark 1
  • Remark 2