Abstract: In this talk, we study the problem of minimizing the distance from a data point to a subvariety of the Grassmannian from the perspective of metric algebraic geometry. An important invariant in this line of work is the number of critical points of this optimization problem for general data. When the data is a general point in the ambient space, this number is called the Euclidean distance (ED) degree of the variety. However, we focus mainly on the case when the data is a general point in the Grassmannian, and call the number of critical points the Grassmann distance (GD) degree. As we will see, the GD degree is more subtle than the ED degree. We give formulas for the ED and GD degrees of prominent subvarieties of the Grassmannian, including Schubert and matroid varieties.

Abstract: The Transformer from "Attention Is All You Need" is now the ubiquitous architecture in state-of-the-art models such as GPT and AlphaFold. At Transformer's heart and soul is the "attention mechanism". We propose an attention-based deep direct sampling method for solving Electrical Impedance Tomography, a class of boundary value inverse problems. We try to provide an example of whether and how one can benefit from the theoretical structure of a mathematical problem to develop task-oriented and structure-conforming neural operators. Then, inspired by the idea of employing partial differential equation (PDE) solvers as preconditioners for this inverse problem, we investigate a "solver-in-the-loop" training mechanism for Transformer-based neural operators. It allows the interaction between learnable parameters in a PDE solver module and those in neural networks for image reconstruction. The new method achieves superior reconstruction quality compared with its predecessors and is robust to noise.

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