A Learnable Proximal Gradient Unrolling Network for Sparse Learning: A Mathematical Optimization–Driven Machine Learning Framework

Sparse learning is a fundamental topic connecting mathematical optimization and machine learning, and it is widely applied in signal reconstruction, feature selection, and robust regression. However, classical iterative solvers for sparse models often require careful manual parameter tuning and may converge slowly under ill-conditioned data or noisy observations. To address these limitations, this study develops a Learnable Proximal Gradient Unrolling Network (LPG-Net) by transforming the iterations of proximal gradient descent into a trainable deep architecture. The proposed method starts from the Least Absolute Shrinkage and Selection Operator (LASSO) formulation and embeds the proximal operator of the ℓ1-regularizer into each network layer, while enabling data-driven adaptation of key algorithmic parameters such as step sizes and thresholding strengths across layers. In addition, a monotonicity-inspired regularization term is introduced to encourage stable descent behavior during training. Experiments on sparse regression and signal denoising tasks indicate that LPG-Net achieves more accurate sparse recovery and faster inference than traditional optimization baselines and standard neural predictors, while retaining strong interpretability due to its explicit connection to optimization updates. The framework provides a principled pathway for integrating mathematical optimization structures into machine learning models for sparse and noise-robust learning problems.