UC Berkeley

The Grassmannian Gr(k, n) is the set of k-dimensional subspaces of the vector space Rn. It is an important object in algebraic geometry, but also in applications, particularly statistics and data science. The Grassmannian is an algebraic variety, and therefore has equational representations. The equational representations one uses in algebraic geometry and in applications are different, and the interplay between these different representations will be an important tool and object of study for us. The equational representation used in algebraic geometry is called the Plücker embedding, and we introduce the projection Grassmannian for the equational representation from applications.Metric algebraic geometry is a relatively new field which bridges algebraic and differential geometry by investigating metric aspects of algebraic varieties. One important aspect of metric algebraic geometry is the study of algebraic optimization: given a data point and a variety, what point on the variety is closest to the data point? We answer this question by computing critical points of the optimization problem. The number of complex critical points is constant for generic data and is called the algebraic degree of the problem.We develop a framework for algebraic optimization and describe the prime ideals of the Grassmannian and related varieties in the first part of the thesis. In Parts II and III, we study optimization problems on Gr(k, n) from an algebraic perspective.In the second part of the thesis, we explore various polynomial optimization problems on the Grassmannian from the perspective of metric algebraic geometry. The multi-eigenvector problem, canonical correlation analysis, and correspondence analysis are all polynomial optimization problems on Grassmannians and fag varieties. Their critical points are all real, and come from the spectral theorem and the singular value decomposition. We then turn to a constrained eigenvalue problem in quantum chemistry, namely the optimization of the Rayleigh quotient on tensor train varieties. We prove that tensor train varieties are parametrized by products of Grassmannians. We compute critical points and data discriminants for this problem. We conclude with a study of Euclidean distance optimization for subvarieties of the Grassmannian with data also in the Grassmannian. This leads us to introduce the Grassmann distance degree as an algebraic measure of the difficulty of this problem. We characterize the Schubert varieties for which this problem is particularly simple.In the third part, we take the perspective of algebraic statistics and study maximum likelihood estimation on a family of probability distributions called determinantal point processes. We prove that, for a dense open subset of determinantal point processes, the maximum likelihood estimation problem can be solved by recursion. We illuminate the relationship between projection determinantal point processes and the Grassmannian via a new variety called the squared Grassmannian sGr(k, n). We prove that, for k =2, all complex critical points of the maximum likelihood estimation problem on the squared Grassmannian are real and positive. We introduce squared linear models and show that they share this remarkable property.