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Geometry In Numerical Algorithms
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Amit Boyarski, Ph.D. Thesis Seminar
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Monday, 11.10.2021, 11:30
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Zoom Lecture: 9804765977
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Advisor:  Prof. Alex M. Bronstein
A common paradigm in engineering consists of problem modeling followed by numerical optimization. Over the years, a chasm has formed between the two stages: Models are becoming more and more complicated in order to address data irregularities while numerical optimization is being delegated to an external solver, usually not designed to handle the specific problem at hand. In this thesis I focus on problems in which there exist some underlying geometric or topological structure. Such structure naturally exists in applications involving shapes where it is explicitly encoded in the form of a discretized manifold. Popular examples include correspondence and reconstruction of non-rigid shapes. In other fields the geometry may be latent but can be inferred from data. For example, in recommender systems it is common to assume a relational structure between different users and different items that can be encoded ex-ante in the form of graphs. Focusing on the above and related examples, I identify a close connection between their underlying geometry and the computational structures arising within common numerical algorithms. Once this connection is established, I show how the problems can be addressed via an appropriate model, often very simple, that can lead to dramatic improvements in many aspects compared to existing literature. These include running times, robustness, accuracy, and ease of implementation.
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