Convolutional Neural Networks (CNNs) are highly effective for image reconstruction problems. Typically, CNNs are trained on large amounts of training images. Recently, however, un-trained neural networks such as the Deep Image Prior and Deep Decoder have achieved excellent image reconstruction performance for standard image reconstruction problems such as image denoising and image inpainting, without using any training data. This success raises the question whether un-trained neural networks can compete with trained ones for practical imaging tasks. To address this question, we consider accelerated magnetic resonance imaging (MRI), an important medical imaging problem, which has received significant attention from the deep-learning community, and for which a dedicated training set exists. We study and optimize un-trained architectures, and as a result, propose a variation of the architectures of the deep image prior and deep decoder. We show that the resulting convolutional decoder out-performs other un-trained methods and---most importantly---achieves on-par performance with a standard trained baseline, the U-net, on the FastMRI dataset, a new dataset for benchmarking deep learning based reconstruction methods. Besides achieving on-par reconstruction performance relative to trained methods, we demonstrate that a key advantage over trained methods is robustness to out-of-distribution examples.

Employing deep neural networks as natural image priors to solve inverse problems either requires large amounts of data to sufficiently train expressive generative models or can succeed with no data via untrained neural networks. However, very few works have considered how to interpolate between these no- to high-data regimes. In particular, how can one use the availability of a small amount of data (even $5-25$ examples) to one's advantage in solving these inverse problems and can a system's performance increase as the amount of data increases as well? In this work, we consider solving linear inverse problems when given a small number of examples of images that are drawn from the same distribution as the image of interest. Comparing to untrained neural networks that use no data, we show how one can pre-train a neural network with a few given examples to improve reconstruction results in compressed sensing and semantic image recovery problems such as colorization. Our approach leads to improved reconstruction as the amount of available data increases and is on par with fully trained generative models, while requiring less than $1 \%$ of the data needed to train a generative model.