Functional Learning: From Theory to Application in Bioimaging
Principal Investigator: Prof. Michael Unser
Participants: Shayan Aziznejad, Thomas Debarre, Jonathan Dong, Sebastian Neumayer, Rahul Parhi, Thanh-An Pham, Mehrsa Pourya
Period: October 1, 2021 — September 30, 2026.
While deep neural networks have shown a remarkable ability to reconstruct biomedical images, they also raise serious reservations that pertain to the stability of these tools and the extent to which one can trust their output. The main concern, which motivates this research program, is that it is very difficult to control their Lipschitz constant. This quantity measures the degree to which a small perturbation of the input can have a huge effect on the output. At worst, a network with a poor Lipschitz constant can be devastating in the context of image reconstruction.
We believe that better Lipschitz constants will result from much shallower neural architectures, 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. In particular, we shall investigate a variational formulation of robust learning that relies on a new HDTV criterion. It 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:
- a "best-of-both-worlds" approach to the reconstruction of biomedical images, 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.