Self-supervised learning from noisy and incomplete data

Many important problems in science and engineering boil down to inferring a signal from noisy and/or incomplete observations, where the observation process is known. Historically, this problem has been tackled using some sort of hand-crafted regularization (sparsity, total variation) to obtain meaningful estimates. Recent data-driven methods often offer better solutions by directly learning a solver from examples of ground-truth signals and associated observations. However, in many real-world applications, ground-truth signals for training are very expensive or even impossible to obtain. Self-supervised learning methods offer a promising alternative by learning a solver from measurement data alone, bypassing the need for ground-truth references. This manuscript provides a comprehensive summary of different self-supervised methods for inverse problems, with a special emphasis on their theoretical underpinnings and presents practical applications in imaging inverse problems.