Abstract: Methods in numerical algebraic geometry have been used to solve a wide range of polynomial systems arising in science and engineering. For example, algebraic kinematics considers the class of kinematic problems arising from polynomial constraints. A common example is synthesizing a linkage that depends on rigid links and rotational joints. For a robot built from sagging cables, the constraints are no longer algebraic. By developing a reformulation of the Irvine sagging cable model, this talk will consider applying monodromy solving to the resulting non-algebraic system aiming to compute all kineto-static equilibrium. Examples from 8-cable spatial cable-driven parallel robots will be used to demonstrate the approach. This is joint work with Aravind Baskar, Mark Plecnik, and Charles Wampler.

Link to article: A numerical continuation approach using monodromy to solve the forward kinematics of cable-driven parallel robots with sagging cables

Abstract: A Brownian motion tree (BMT) model is a Gaussian model whose associated set of covariance matrices is linearly constrained according to common ancestry in a fixed phylogenetic tree. We study the complexity of inferring the maximum likelihood (ML) estimator for a BMT model by computing its ML-degree. Our main result is that the ML degree of the BMT model on a star tree with n+1 leaves is 2^(n+1)-2n-3, which was previously conjectured by Amendola and Zwiernik. This talk will focus on a combinatorial formula for the determinant of the concentration matrix of a BMT model, which generalizes the Cayley-Prufer theorem to complete graphs with weights given by a tree, and on the intersection theory used to compute the ML-degree of the star tree.

Abstract: The topic of Economic Fragility was seen to be important first hand as the global economy witnessed shocks to the supply chain during the covid-19 lockdown era. Ben Golub, Matt Elliot and M. V. Leduc in recent work proposed a model to begin to explain why shocks to the economy can lead to disruption of delivery of goods. Their work found an expected phase transition. In this joint work with Jose Israel Rodriguez, a characterization of the transition point is given in terms of a polynomial system and a discriminant. With this algebraic framework we are able to study multiparameter versions of supply chain models, which brings us closer to modeling supply chains in real life scenarios.

Abstract: The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having integer or rational coefficients, the solution set is well-defined. When solving parameterized polynomial systems with parameter values that are not exact due to imprecision in measurement or prior computations, the structure of the solution set can change. This talk will describe methods to robustly recover nearby parameters corresponding to the desired structure for solutions at infinity, positive dimensional components, multiplicity, and irreducible components. These methods will be demonstrated through illustrative examples and problems arising from the kinematics of mechanisms and robots.

Abstract: This talk will define a Markov basis and then evaluate the challenges and best practices associated with the Markov bases approach to sampling from conditional distributions. We provide insights and clarifications after 25 years of the publication of the fundamental theorem for Markov bases by Diaconis and Sturmfels. We also prove three new results on the complexity of Markov bases in hierarchical models, relaxations of the fibers in log-linear models, and limitations of partial sets of moves in providing an irreducible Markov chain. This short talk will give an overview of these results and their implications in practice.

Abstract: Motivated by previous work on moment varieties for Gaussian distributions and their mixtures, we study moment varieties for two other statistically important two-parameter distributions: the inverse Gaussian and gamma distributions. In this talk, we will show how techniques from commutative algebra and algebraic geometry can be used to realize these moment varieties as determinantal varieties. This allows us to provide evidence for algebraic identifiability of mixtures of inverse Gaussian and mixtures of gamma distributions. This is joint work with Oskar Henriksson and Lisa Seccia.

arXiv:2312.10433

Abstract: Many problems in computer vision are represented using a parameterized polynomial system, where the solution set is critical for 3D reconstruction. Minimal problems can be of special interest as these polynomial systems are well-constrained and generically have finitely many solutions. Computing the Galois/monodromy group of the associated branch cover can yield information and understanding about the underlying structure of the minimal problem. One potential outcome of this computation is identifying the existence of possible decompositions or symmetries. Beyond the question of existence, one would like to compute formulas for these symmetries, towards the eventual goal of solving the systems more efficiently. These Galois/monodromy groups can be computed using numerical homotopy continuation via a multitude of softwares. The equations for the symmetries can be found with the additional technique of multivariate rational function interpolation. This talk will discuss these methods in theory, as well as illustrate the approach on practical examples of minimal problems in computer vision. This is joint work with Timothy Duff, Viktor Korotynskiy, and Tomas Pajdla.

Abstract: The representation of evolutionary relationships using phylogenetic trees has been central to understanding biological processes, such as speciation, the spread of pathogens, and cancer evolution. One of the key challenges in phylogenetics is that gene trees, which represent the evolutionary relationships of a single genetic locus across a set of individuals, often exhibit topological disagreements, both among themselves and with the species tree, which describes the evolutionary relationships among the populations or species from which these individuals are drawn. Recent and continuing rapid advances in genomic sequencing technologies have produced large-scale genomic datasets, revealing that gene tree-species tree incongruence is widespread. We develop a new method to reliably infer species trees from gene trees despite the incongruences using incomplete u-statistics. Because the incomplete u-statistics method specializes in performing effective hypothesis testing when a model has singular points, we apply the method to test the Multi-Species Network Coalescent Model (MSNC)—one of the most commonly used models described by algebraic constraints. We have shown that the incomplete u-statistics method verifies the behavior of the MSNC model at irregular points in all four of its submodels, which was first suggested by the MSCQuartets method—a method that does not adequately address the singular points in irregular models. Our next goal is to generalize the method to other models, such as the Cavender-Farris-Neyman (CFN) model, and apply it to real biological data.

Abstract: The Newton-Okounkov body is a convex set which generalizes the Newton polytope. In applications, the volume of the Newton-Okounkov body has been used to bound the number of solutions to zero-dimensional systems of equations. In this talk, we will see how to compute the Newton-Okounkov body using subalgebra (SAGBI) bases (as opposed to Khovanskii bases). This presents a more effective computation of the volumes of these bodies. In the process of doing this, we will prove a strong equivalence between subalgebra bases and Khovanskii bases.

Abstract: We examine two different notions of tropical positivity for symmetric determinantal varieties of certain ranks. In particular, following the language of Brandenburg, Loho, and Sinn, we describe the positive part and the really positive part of the tropical variety of symmetric tropical rank 2 matrices, and of the tropical hypersurface of rank-deficient symmetric tropical matrices. We show that these two notions of positivity coincide for the rank 2 case, but do not for the hypersurface.

Abstract: In this work, we implement a Reinforcement Learning algorithm to sample integer points in a polytope. The polytope arises as a fiber of a log-linear model. More specifically, we develop a novel Progressive Actor-Critic algorithm and prove its convergence to a approximately optimal fiber sampling policy. The progression algorithm is used to bypass a common bottleneck of computation of Markov basis.