We study multivariate Gaussian statistical models whose maximum likelihood estimator (MLE) is a rational function of the observed data. We establish a one-to-one correspondence between such models and the solutions to a nonlinear first-order partial differential equation (PDE). Using our correspondence, we reinterpret familiar classes of models with rational MLE, such as directed (and decomposable undirected) Gaussian graphical models. We also find new models with rational MLE. For linear concentration models with rational MLE, we show that homaloidal polynomials from birational geometry lead to solutions to the PDE. We thus shed light on the problem of classifying Gaussian models with rational MLE by relating it to the open problem in birational geometry of classifying homaloidal polynomials.

We explore the maximum likelihood degree of a homogeneous polynomial F on a projective variety X, MLDF(X), which generalizes the concept of Gaussian maximum likelihood degree. We show that MLDF(X) is equal to the count of critical points of a rational function on X and give different geometric characterizations of it via topological Euler characteristic, dual varieties, and Chern classes.

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.

Denna avhandling utforskar optimeringsutmaningar inom algebraisk statistik genom att använda både topologiska och geometriska metoder för att nå nya insikter. Kärnuppdraget innefattar att avgöra de förväntade antalet komplexa kritiska punkter till algebraiska optimeringsproblem med bivillkor. Detta utgör en grund för att förstå optimering av log-likelihood funktionen för multivariata normalfördelningar och Euklidiska avståndsfunktionen. Det förväntade antalet lösningar till ett optimeringsproblem med bivillkor är mer allmänt känt som en optimeringsgrad. Optimeringsgraden kan användas för numeriska lösningsmetoder då den räknar antalet lösningar utan att explicit beräkna lösningarna. Optimeringsgraden kompletteras av en parallell forskningsbana som kompletterar och expanderar de primära temana genom att studera det omvända problemet när det finns en kritisk punkt inuti ett speciellt delområde och att förstå multivisa varieteten för linjer i bildkorrigeringssyfte. 

I artikel A utforskas linjära statistiska koncentrationsmodeller av centrerade multivariata gaussiska slumpvariabler. Vårt fokus ligger på att beräkna antalet kritiska punkter för log-likelihoodfunktionen inom ett linjärt rum av symmetriska matriser. Artikeln bidrar med nya formler för Gaussisk maximal sannolikhetsgrad, hämtade från linjegeometri och Segre-klasser i snittteori. Vi tar även upp modeller med codimension ett och scenarier med noll kritiska punkter.

I Artikel B fördjupar vi undersökningen från Artikel A genom att utforska gaussisk likelihood-geometri hos godtyckliga projektiva varieteter. Vi introducerar även maximum likelihoodgraden för ett homogent polynom över en projektiv varietet och går djupare in på att kvantifiera de kritiska punkterna för en rationell funktion. Det hittas olika tillvägagångssätt för att beräkna maximum likelihoodgraden via geometriska karakteriseringarna såsom Euler karakteristik, duala varieteter och Chern-klasser.

Artikel C främjar undersökningen av multivariata gaussiska statistiska modeller med rationella maximum-likelihood-estimator (MLE). En korrespondens etableras mellan dessa modeller och lösningar till en icke-linjär partiell differentialekvation av första ordningen (PDE). Denna koppling belyser klassificeringen av gaussiska modeller med rationell MLE och relaterar den till det öppna problemet inom birationell geometri som omfattar klassificeringen av homaloida polynom.

Artikel D beräknar den generiska maximum likelihoodgraden för en varietet, som en analog till den kända formeln med polära klasser för den euklidiska avståndsgraden. Dessutom klassificeras alla projektiva kurvor av maximum likelihoodgrad 1, som öppnar upp för frågan om de går att realisera som reella statistiska modeller.

Artikel E korsar algebraisk geometri och datorseende, med fokus på projicerade linjer från flera pinhole-kameror. Linjernas multivisa varietet fångar dessa projektioner som en algebraisk varietet. Huvudresultatet fastställer idealet för denna varietet, genererat av 3x3-minorer av en matris som härleds från ekvationerna för de projicerade linjerna. Föregångaren till linje-multivisa varieteten är den punkt-multivisa varieteten, där korrigering av bilder var en drivande motivation för att introducera den Euklidiska avståndsgraden. Märkbart, så kopplar Artikel F den nya multivisa varieteten till begreppet av Euklidisk avståndsgrad och öppnar dörren för att studera linjemultivisa varieteten inom kontexten för 3D rekonstruktion.

I Papper F fördjupade vi oss i begreppet Euklidiska avståndsuppskattningar inom ramen för en specifik delmängd av tillgängliga data. För att konstruera en robust grundteori introducerar detta papper begreppen 'relativ dualitet' och 'relativa karakteristiska klasser.' Det visar att klassiska formler kan uttryckas ekvivalent i en relativ formulering och därmed belysa de geometriska komplexiteter som är inneboende i relativ analys.

We study multivariate Gaussian models that are described by linear conditions on the concentration matrix. We compute the maximum likelihood (ML) degrees of these models. That is, we count the critical points of the likelihood function over a linear space of symmetric matrices. We obtain new formulae for the ML degree, one via line geometry, and another using Segre classes from intersection theory. We settle the case of codimension one models, and characterize the degenerate case when the ML degree is zero.

We introduce a theory of relative tangency for projective algebraic varieties. The dual variety X^∨_Z of a variety X relative to a subvariety Z is the set of hyperplanes tangent to X at a point of Z. We also introduce the concept of polar classes of X relative to Z. We explore the duality of varieties of low rank matrices relative to special linear sections. In this framework, we study the critical points of the Euclidean Distance function from a data point to X, lying on Z. The locus where the number of such conditional critical points is positive is called the ED data locus of X given Z. The generic number of such critical points defines the conditional ED degree of X given Z.We show the irreducibility of ED data loci, and we compute their dimensions and degrees in terms of relative characteristic classes

We study multivariate Gaussian statistical models whose maximum likelihood estimator (MLE) is a rational function of the observed data. We establish a one-to-one correspondence between such models and the solutions to a nonlinear first-order partial differential equation (PDE). Using our correspondence, we reinterpret familiar classes of models with rational MLE, such as directed (and decomposable undirected) Gaussian graphical models. We also find new models with rational MLE. For linear concentration models with rational MLE, we show that homaloidal polynomials from birational geometry lead to solutions to the PDE. We thus shed light on the problem of classifying Gaussian models with rational MLE by relating it to the open problem in birational geometry of classifying homaloidal polynomials.

We explore the maximum likelihood degree of a homogeneous polynomial F on a projective variety X, MLD_F(X), which generalizes the concept of Gaussian maximum likelihood degree. We show that MLD_F(X) is equal to the count of critical points of a rational function on X, and give different geometric characterizations of it via topological Euler characteristic, dual varieties, and Chern classes.

We study the following problem in computer vision from the perspective of algebraic geometry: Using m pinhole cameras we take m pictures of a line in P^3. This produces m lines in P² and the question is which m-tuples of lines can arise that way. We are interested in polynomial equations and therefore study the complex Zariski closure of all such tuples of lines. The resulting algebraic variety is a subvariety of (P²)^m and is called line multiview variety. In this article, we study its ideal. We show that for generic cameras the ideal is generated by 3×3-minors of a specific matrix. We also compute Gröbner bases and discuss to what extent our results carry over to the non-generic case.

We introduce the F-adjoined Gauss map. We use it to express the Gaussian maximum likelihood degree as a product of two invariants. As an application of our product formula, we classify all projective curves of Gaussian maximum likelihood degree 1. We also provide a formula for the generic Gaussian maximum likelihood degree of a projective variety X in terms of its polar classes. The renowned polar class formula for generic Euclidean distance degree is a special case of our formula.