We present PROSUB: PROgressive SUBsampling, a deep learning based, automated methodology that subsamples an oversampled data set (e.g. channels of multi-channeled 3D images) with minimal loss of information. We build upon a state-of-the-art dual-network approach that won the MICCAI MUlti-DIffusion (MUDI) quantitative MRI (qMRI) measurement sampling-reconstruction challenge, but suffers from deep learning training instability, by subsampling with a hard decision boundary. PROSUB uses the paradigm of recursive feature elimination (RFE) and progressively subsamples measurements during deep learning training, improving optimization stability. PROSUB also integrates a neural architecture search (NAS) paradigm, allowing the network architecture hyperparameters to respond to the subsampling process. We show PROSUB outperforms the winner of the MUDI MICCAI challenge, producing large improvements > 18% MSE on the MUDI challenge sub-tasks and qualitative improvements on downstream processes useful for clinical applications. We also show the benefits of incorporating NAS and analyze the effect of PROSUB’s components. As our method generalizes beyond MRI measurement selection-reconstruction, to problems that subsample and reconstruct multi-channeled data, our code is [7].