General Decomposition Pursuit Algorithm for Linear Inverse Problems

Over the past few decades, various numerical methods have been developed to solve linear inverse problems with sparse solutions. However, there remains a shortage of efficient algorithms specifically tailored for large-scale problems. In this paper, we work toward bridging this gap by developing the so-called General Decomposition Pursuit algorithm, which is designed to directly tackle a large-scale problem. This algorithmic framework is notably distinct from any existing sparsity-aware method. The main mechanism of the algorithm involves decomposing a large problem into several mutually coupled small subproblems and then combining the (inexact) solutions of these subproblems to generate a sparse solution to the original problem. The global convergence of the algorithm is shown under the restricted isometry property. Simulations with synthetic data and applications in medical image reconstruction indicate that the proposed algorithm, when equipped with an appropriate inner solver for subproblems, can outperform several mainstream algorithms in success rates for locating the sparse solution to the problem.