Kubjas, Kaie, Kolokvij marec 2019

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One of many definitions gives the rank of an m times n matrix M as the smallest natural number r such that M can be factorized as AB, where A and B are m times r and r times n matrices respectively. In many applications, we are interested in factorizations of a particular form. For example, factorizations with nonnegative entries define the nonnegative rank and are closely related to mixture models in statistics.

Nonnegative and psd rank have geometric characterizations using nested polytopes. I will explain how to use these characterizations to study the semialgebraic geometry of the set of matrices of given nonnegative rank. In particular, I will recall what was previously known about the set of matrices of nonnegative rank at most $r$ for r=1,2,3 and present recent results on the boundaries of the set of matrices of nonnegative rank at most four using notions from the rigidity theory of frameworks. These results are based on joint work with Robert Krone.

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