The topic of this workshop is solving systems of polynomial equations using numerical algorithms. The participants will learn to work with the software HomotopyContinuation.jl, so that they can use it for their own research problems. In addition, the workshop will feature a series of lectures on the theory of numerical homotopy continuation, as well as contributed talks by Timothy Duff, Alex Heaton, Julia Lindberg and Maggie Regan.
Many problems in applications such as computer vision and engineering can be formulated as solving a parameterized system of polynomial equations for various instances of the parameters. By using homotopy continuation (Paul’s lecture) within numerical algebraic geometry, one can solve these parameterized polynomial systems using parameter homotopies (Sascha’s lecture). In computer vision, parameter homotopies can be naturally overdetermined which can lead to challenges when performing numerical computations. It can also be worthwhile to construct real parameter homotopies in order to more efficiently compute the real solutions for the particular application. This talk will use examples to discuss methods to overcome various challenges as well as discuss methods for constructing parameter homotopies that only compute real solutions.
Using both Monodromy (Taylor's lecture) and Parameter Homotopies (Sascha's lecture) we will apply Numerical Nonlinear Algebra to statistics. Usually, a Voronoi cell is the subset of points closest to your favorite point, as measured by Euclidean distance. If your favorite point lies on a statistical model, the log-likelihood function (of maximum likelihood estimation) can replace Euclidean distance, and the resulting Voronoi cells are called logarithmic. All points in a logarithmic Voronoi cell have the same maximum likelihood estimate on the statistical model. We will use Numerical Nonlinear Algebra to compute the logarithmic Voronoi cells of our favorite point on a statistical model, and observe directly their (sometimes) nonlinear boundaries.
This is a talk about symmetries in polynomial systems. Once we know what they look like, we can exploit them in various ways (e.g. inside of parameter homotopies, monodromy, etc.) How do we know if they exist? One reasonable answer comes via the monodromy group itself -- viewed as a subgroup of the symmetric group, the centralizer of the monodromy group is isomorphic to a group of rational deck transformations. These deck transformations are typically the symmetries we seek. In general, the equations defining them might be complicated. A notable example of such a deck transformation is the "twisted pair" map which appears when estimating the relative pose between two cameras. Two solutions to this problem which differ up to a twisted pair map to the same essential matrix. The ideal of polynomials that vanish on all essential matrices is minimally generated by ten cubic equations. I will introduce an analogous problem involving 3 cameras and the interesting equations we discovered. (Joint work with Viktor Korotynskiy, Tomas Pajdla, and Maggie Regan.)
The method of moments is a statistical method for density estimation that equates sample moments to moment equations for a given family of densities. When the underlying distribution is assumed to be a convex combination of Gaussian densities, the resulting moment equations are polynomial in the density parameters. Using monodromy, polyhedral and parameter homotopy methods, we examine the asymptotic behavior of the variety stemming from these equations as the number of components and the dimension of each component increases. This is joint work with Jose Israel Rodriguez and Carlos Amendola.
