Recently, with the enormous development of deep learning techniques, solving underdetermined linear systems (more unknowns than equations) have become one of major concerns in medical imaging. Typical examples include undersampled MRI, local tomography, and sparse view CT, where deep learning techniques have shown excellent performance. Although deep learning methods appear to overcome limitations of existing mathematical methods in handling various underdetermined problems, there is a lack of rigorous mathematical foundations which would allow us to understand reasons why deep learning methods perform that well. This talk deals with this learning causal relationship about structure of training data suitable for deep learning to solve highly underdetermined inverse problems. We examine whether or not a desired reconstruction map can be learnable from the training data and the underdetermined system. Most problems of solving underdetermined linear systems in medical imaging are highly non-linear.
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