Phase Transition in Binary Compressed Sensing via Annealing with Adaptive Regularization

Compressive sensing (CS) is a signal processing technique used to reconstruct sparse signals from significantly fewer measurements than traditionally required. While CS has been extensively studied for continuous signals, many practical applications in digital communications and sensor networks involve signals that are inherently binary. This specialization, known as binary compressed sensing (BCS), has recently been reformulated as a quadratic unconstrained binary optimization (QUBO) problem to leverage the power of annealing-based optimizers. However, the reconstruction performance of QUBO-based BCS is highly sensitive to the regularization parameter, and a principled selection strategy remains an open challenge. In this work, we demonstrate that using a fixed regularization parameter fails to produce consistent phase transition behavior across different sampling and sparsity regimes. By revealing that the optimal parameter follows a structured, instance-dependent pattern, we propose a learning-based framework to predict near-optimal values directly from problem characteristics. Numerical experiments show that our adaptive regularization stabilizes phase transitions under simulated annealing and transfers effectively to quantum annealing solvers. Binary image reconstruction experiments further confirm its practical advantages over fixed-parameter baselines, providing a scalable pathway toward reliable annealing-based signal reconstruction.