Randomized block coordinate descent method for linear ill-posed problems

TL;DR

This work develops and analyzes a randomized block coordinate descent method for linear ill-posed problems in separable form $\sum_{i=1}^{b} A_i x_i = y$, addressing large-scale, memory-constrained settings with noisy data. The authors establish stability, weak convergence, and discrepancy-principle-based stopping (a priori and a posteriori), and prove strong convergence in a tensor-product forward model, demonstrating the method's regularization properties. They extend the approach to include separable convex regularization via a Bregman-distance framework, yielding the RBCD+ algorithm with provable convergence. Numerical experiments in computed tomography and coded aperture temporal imaging validate the theory, showing competitive accuracy and finite-time termination, and highlighting the practical impact for high-dimensional inverse problems.

Abstract

Consider the linear ill-posed problems of the form $\sum_{i=1}^{b} A_i x_i =y$, where, for each $i$, $A_i$ is a bounded linear operator between two Hilbert spaces $X_i$ and ${\mathcal Y}$. When $b$ is huge, solving the problem by an iterative method using the full gradient at each iteration step is both time-consuming and memory insufficient. Although randomized block coordinate decent (RBCD) method has been shown to be an efficient method for well-posed large-scale optimization problems with a small amount of memory, there still lacks a convergence analysis on the RBCD method for solving ill-posed problems. In this paper, we investigate the convergence property of the RBCD method with noisy data under either {\it a priori} or {\it a posteriori} stopping rules. We prove that the RBCD method combined with an {\it a priori} stopping rule yields a sequence that converges weakly to a solution of the problem almost surely. We also consider the early stopping of the RBCD method and demonstrate that the discrepancy principle can terminate the iteration after finite many steps almost surely. For a class of ill-posed problems with special tensor product form, we obtain strong convergence results on the RBCD method. Furthermore, we consider incorporating the convex regularization terms into the RBCD method to enhance the detection of solution features. To illustrate the theory and the performance of the method, numerical simulations from the imaging modalities in computed tomography and compressive temporal imaging are reported.

Figures (11)

• Figure 1: The Sheep-Logan phantom (left) and a real CT image (right).
• Figure 2: The reconstruction results by the RBCD method (left) and CBCD method (middle), and the numerical reconstruction errors of two methods (right) from the exact full angle tomographic data of the Shepp-Logan phantom, respectively.
• Figure 3: The reconstruction results of Algorithm \ref{['alg:RBCD_DP']} from the sparse view tomographic data with 60 projection angles and three relative noise levels $\delta_{\text{rel}}=0.01$ (left), $0.02$ (middle), and $0.03$ (right) of the Shepp-Logan phantom (top) and real chest CT image (bottom).
• Figure 4: The reconstruction results of Algorithm \ref{['alg:RBCD_DP']} from the limited angle tomographic data with 160 projection angles within $[10^\circ,170^\circ)$ with $1^\circ$ steps and three relative noise levels $\delta_{\text{rel}}=0.01$ (left), $0.02$ (middle), and $0.03$ (right) of the Shepp-Logan phantom (top) and real chest CT image (bottom).
• Figure 5: The original video of the dataset Runner.

Theorems & Definitions (28)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• Theorem 2.4
• proof
• Remark 2.5
• Theorem 2.6
• proof