This research program is motivated by the remarkable ability of deep neural networks to improve the quality of biomedical image reconstruction. While the results reported so far are extremely encouraging, serious reservations have been voiced pertaining to the stability of these tools and the extent to which we can trust their output. The main concern is that it is very difficult to control the Lipschitz constant of the current neural architectures. This means that a small perturbation of the input can result in a huge deviation of the output, which can have devastating effects in the context of image reconstruction. We believe that the remedy lies in the use of much shallower networks, which are easier to control. However, a reduction in the number of layers will degrade the performance, unless we augment the sophistication of the primary modules; in particular, the nonlinear ones. By drawing on our career-long experience with splines, we therefore propose to rely on the powerful tools of functional optimization to improve learning architectures. This will allow us to develop two novel approaches to learning: sparse simplicial splines, and hierarchical spline networks—an extension of the popular deep ReLU neural networks In parallel, we shall develop specific neural networks to solve two outstanding problems in biomedical imaging: - A “best-of-both-worlds” approach to biomedical image reconstruction, involving the stable integration of state-of-the-art physics-based solvers with the new tools of machine learning; - The 3D reconstruction of the entire manifold of configurations of a biomolecule from a large collection of very low-dose cryo-electron tomograms. This goal, which may be viewed as the Graal of structural biology, has remained elusive so far and calls for an entirely new paradigm for single-particle analysis.
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