B-SMART: Bregman-Based First-Order Algorithms for Non-negative Compressed Sensing Problems

We introduce and study Bregman functions as objectives for non-negative sparse compressed sensing problems together with a related first-order iterative scheme employing non-quadratic proximal terms. This scheme yields closed-form multiplicative updates and handles constraints implicitly. Its analysis does not rely on global Lipschitz continuity in contrast to established state-of-the-art gradient-based methods, hence it is attractive for dealing with very large systems. Convergence and a O(k − 1) rate are proved. We also introduce an iterative two-step extension of the update scheme that accelerates convergence. Comparative numerical experiments for non-negativity and box constraints provide evidence for a O(k − 2) rate and reveal competitive and also superior performance.