We present a brief overview of computational image reconstruction methods that assume that Magnetic Resonance Imaging (MRI) data possesses shift-invariant autoregressive characteristics, where the unique autoregressive structure of each dataset is learned from a small amount of scan-specific calibration data. Our discussion focuses particular attention on a method we recently introduced named LORAKI. LORAKI is a learning-based image reconstruction method that relies on scan-specific nonlinear autoregressive modeling using a recurrent convolutional neural network, and has demonstrated better performance than previous approaches. As a novel contribution, we also describe and evaluate an extension of LORAKI that makes simultaneous use of support, phase, parallel imaging, and sparsity constraints, where the balance between these different constraints is automatically determined through the training procedure. Results with real data demonstrate that this modification leads to further performance improvements.