Neural Approaches for Geometric Problems

Machine learning, especially the use of neural netowrks have shown great success in a broad range of applications. Recently, neural approaches have also shown promise in tackling (combinatorial) optimization problems in a data-driven manner. On the other hand, for many problems, especially geometric optimization problems, many beautiful geometric ideas and algorithmic insights have been developed in fields such as theoretical computer science and computational geometry. Our goal is to infuse geometric and algorithmic ideas to the design of neural frameworks so that they can be more effective and generalize better. In this talk, I will give two examples in this direction. The first one is what we call a mixed Neural-algorithmic framework for the Steiner Tree problem in the Euclidean space, leveraging the celebrated PTAS algorithm by Arora. Interestingly, here the model complexity can be made independent of the input point set size. The second one is an neural architecture for approximating the Wasserstein distance between point sets, whose design /analysis uses a geometric coreset idea.