Improved analytical learning proximal operator method for sparse recovery

•Proposing an improved ALePOM (I-ALePOM) method for sparse recovery, which has a higher performance than existing deep unfolding methods.•Introducing a long short-term memory (LSTM) cell into each layer of ALePOM, helping to adaptively compute the thresholds and step-sizes for each signal.•Extending the proposed I-ALePOM method from the one-dimensional vector form to the two-dimensional matrix form with theoretical analysis, reducing the computational burden for the sparse matrix recovery problem.•Various numerical experiments are conducted to validate the performance and advantage of the proposed I-ALePOM for different sparse recovery applications.