Deep learning (DL) has emerged as a powerful tool for improving the reconstruction quality of accelerated MRI. These methods usually show enhanced performance compared to conventional methods, such as compressed sensing (CS) and parallel imaging. However, in most scenarios, CS is implemented with two or three empirically-tuned hyperparameters, while a plethora of advanced data science tools are used in DL. In this work, we revisit ℓ1 -wavelet CS for accelerated MRI using modern data science tools. By using tools like algorithm unrolling and end-to-end training with stochastic gradient descent over large databases that DL algorithms utilize, and combining these with conventional concepts like wavelet sub-band processing and reweighted ℓ1 minimization, we show that ℓ1-wavelet CS can be fine-tuned to a level comparable to DL methods. While DL uses hundreds of thousands of parameters, the proposed optimized ℓ1-wavelet CS with sub-band training and reweighting uses only 128 parameters, and employs a fully-explainable convex reconstruction model.