Topics in projective algebraic optimization

Abstract: This thesis explores optimization challenges within algebraic statistics, employing both topological and geometrical methodologies to derive new insights. The main focus is the optimization degree of nearest point and Gaussian maximum likelihood estimation problems with algebraic constraints. The optimization degree counts the number of complex critical points for an optimization problem. It is interesting as it can aid numerical solvers by providing an upper bound on the number of solutions to a set of equations, without computing them explicitly. The study extends to a parallel research trajectory, complementing and expanding the primary themes by studying relative tangency for critical point loci and characterizing the ideal of the line-multiview variety, inspiring further study of reconstructing 3D objects from 2D images in computer vision. Paper A focuses on linear concentration models and critical point counts for the Gaussian log-likelihood function when restricted to a linear space. The paper unveils new Gaussian maximum likelihood degree formulae from line geometry and Segre classes. We also study codimension one models and scenarios with zero maximum likelihood degree in particular.In Paper B, we extend the inquiry from Paper A by exploring Gaussian likelihood geometry of arbitrary projective varieties. We introduce the maximum likelihood degree of a homogeneous polynomial on a projective variety, delving into quantifying critical points for a rational function. We find geometric characterizations of the maximum likelihood degree in terms of Euler characteristics, dual varieties, and Chern classes.Paper C advances the investigation into multivariate Gaussian statistical models with rational maximum likelihood estimator (MLE). A correspondence is established between such models and solutions to a nonlinear first-order partial differential equation (PDE). This link sheds light on the problem of classifying Gaussian models with rational MLE, relating it to the open problem of classification of homaloidal polynomials in birational geometry.Paper D computes the generic, or expected, maximum likelihood degree of a variety as an analog to the known polar class formula for the Euclidean distance degree. Additionally, as a follow-up to paper C, the complex projective curves of maximum likelihood degree 1 are classified in paper D. This allows further work into when a complex curve can be realized as a real statistical models. Both paper C and D connect the maximum likelihood degree as a possible generalization to the Euclidean distance degree for projective varieties.Paper E intersects algebraic geometry and computer vision, focusing on projected lines from multiple pinhole cameras. The line multiview variety captures these projections as an algebraic variety. The main result establishes the ideal of this variety, generated by 3x3-minors of a matrix derived from projected line equations. The predecessor of the line-multiview variety is the point-multiview variety, with image correction being a driving motivation for introducing the Euclidean distance degree. Notably, Paper E opens the door for studying the Euclidean distance degree of the line-multiview variety and its uses in 3D reconstruction.Paper F delves into the concept of Euclidean distance estimates within the context of a specific subset of the available data. To contruct a robust foundational theory, this paper introduces the concepts of relative duality and relative characteristic classes. It demonstrates that classical formulas can be equivalently expressed in the relative setting, thereby shedding light on the geometric intricacies inherent to this relative analysis.