Dissertation Defense | Tensors: An Adaptive Approximation Algorithm, Convergence in Direction, and Connectedness Properties, Feb. 27

Nathaniel McClatchey

Mathematics student Nathaniel McClatchey defends his Ph.D. dissertation on Feb. 27 from 12:30 to 2:30 in Morton 320.

The of his dissertation is “Tensors: An Adaptive Approximation Algorithm, Convergence in Direction, and Connectedness Properties.”

Abstract: Low-rank approximation of tensors is plagued by slow convergence of the sequences produced by popular algorithms such as Alternating Least Squares (ALS), by ill-posed approximation problems which cause divergent sequences, and by poor understanding of the nature of low-rank tensors. First, I develop a novel adaptive approximation method based on ALS. I provide experimental evidence that the adaptive method outperforms ALS. Second, I examine the behavior of sequences produced when optimizing bounded multivariate rational functions. The resulting theorems provide insight into the behavior of certain divergent sequences. Finally, to improve understanding of the nature of low-rank tensors, I examine path and simple connectedness properties of spaces of low-rank tensors.