Why do we regularise in every iteration for imaging inverse problems?

TL;DR

This work explores for the first time the efficacy of ProxSkip to a variety of imaging inverse problems and proposes a novel PDHGSkip version, which has the potential to accelerate computations while maintaining high-quality reconstructions.

Abstract

Regularisation is commonly used in iterative methods for solving imaging inverse problems. Many algorithms involve the evaluation of the proximal operator of the regularisation term in every iteration, leading to a significant computational overhead since such evaluation can be costly. In this context, the ProxSkip algorithm, recently proposed for federated learning purposes, emerges as an solution. It randomly skips regularisation steps, reducing the computational time of an iterative algorithm without affecting its convergence. Here we explore for the first time the efficacy of ProxSkip to a variety of imaging inverse problems and we also propose a novel PDHGSkip version. Extensive numerical results highlight the potential of these methods to accelerate computations while maintaining high-quality reconstructions.

Figures (10)

• Figure 1: Left to right: Ground truth $\bm{u}^{\dagger}\in\mathbb{R}^{200\times300}$. Noisy image $\bm{b}$, $\sigma=0.05$. Dual-ROF $u^{*}$ with $\alpha=0.5$. Dual-Huber-ROF $u^{*}$ (see Section \ref{['sec:huber']}) with $\alpha=0.55$, $\varepsilon=0.1$. The parameters $\alpha$ are optimised with respect to SSIM.
• Figure 2: Top: Comparison of ProjGD and ProxSkip for multiple values of $p$ for \ref{['eq:dual_TV']} with respect to iterations (left) and CPU time (right). Bottom: Detailed versions for the first 100 (left) and the last 100 iterations (right) when $p=0.1$. The vertical dotted lines indicate the iterations where $\mathcal{P}_{C}(\cdot)$ is applied.
• Figure 3: ProjGD, AProjGD and ProxSkip with optimal $p$ value for the \ref{['eq:dual_huber_rof']} problem with respect to iterations (left) and CPU time (right).
• Figure 4: Left to right: Noisy and blurry image. TV deblurred image with $\alpha=0.025$. Difference $|\bm{u}_{k}-\bm{u}^{\ast}|$ for FISTA when it is less than $\varepsilon=10^{-3}$ and $10^{-5}$.
• Figure 5: Comparing ISTA, FISTA and Proxskip for multiple values of $p$ for TV deblurring. The proximal of TV is solved using AProjGD using 10 and 100 iterations. ProxSkip outperforms FISTA when $p = 0.05$ and $0.1$.