Posters

Tim Duff, Cvetelina Hill, and Kisun Lee

Title: Monodromy Solvers

Abstract: We present an algorithmic framework for polynomial system solving based on monodromy actions. Our framework supplements existing coefficient-parameter homotopy techniques with a randomized procedure for discovering solutions that avoids an expensive "initial solve" in cases where the root count differs from that of commonly used start systems. We are free to use various heuristics for finding these solutions and organize those found in a decorated graph. We provide a basic software implementation and observe, based on experiments and probabilistic argument, that our methods scale well with the number of solutions. In current work, we consider schemes for parallelization, the effect of path-tracking failures, and other probabilistic analysis related to the performance of our methods.

Marc Härkönen

Title: Holonomic least angle regression

Abstract: One of the main problems studied in statistics is the fitting of models. Generalized linear models are parametric models in the exponential family, where each parameter corresponds to one covariate. One might then ask which of the covariates are most ``impactful'' in our fitting procedure, and how do should decide which covariates to include in the model. There have been numerous attempts at automatizing this process. Most notably, the Least Angle Regression algorithm, or LARS, is a computationally efficient algorithm that ranks the covariates of a linear model by using Riemannian geometry. We will consider an extension of the LARS algorithm to a class of distributions in the generalized linear model by using the geometry of the manifold of exponential families. This extension is also computationally efficient as long as the normalizing constant is “easy" to compute. However this may not be the case, for example the normalizing constant may contain a complicated integral that has to be evaluated numerically.

In this poster, we will be focusing on the case where the normalizing constant satisfies a holonomic system. We present a modification of the holonomic gradient method that allows the computation of projections and changes on coordinates on the manifold without using numerical integration. We implement this method into the extended LARS algorithm, which we call the holonomic extended least angle regression algorithm, or HELARS."

Aida Maraj

Title: Hierarchical Models and their ​Toric Ideals

Abstract: This poster will be about hierarchical models in Algebraic Statistics. I will present an example where these models apply and introduce their corresponding toric ideals. A formula for calculating Krull dimension will be given. To describe generating sets of these ideals one can use a symmetric group action. Using this tool, we will describe generating sets for some non-reducible models. This is joint work with Uwe Nagel. ​

Daniel Mckenzie

Title: Many k-neighborly polytopes from quivers.

Abstract: A d dimensional polytope on n vertices, P, is said to be k-neighborly if every set of k vertices spans a face of P. Such polytopes provide interesting extremal objects in discrete geometry, and are also useful in the signal processing field of Compressive Sensing. Donoho and Tanner have shown that, in some sense, most polytopes are k-neighborly, although explicitly constructing k-neighborly polytopes remains challenging. In this poster we detail recent work, joint with Patricio Gallardo, on how to construct neighborly polytopes from quivers, where the key step is to verify that the toric variety associated to the polytope satisfies a certain property.

Margaret Regan

Title: Software for computing topology of smooth real surfaces

Abstract: A common computational problem is to compute topological information about a real surface defined by a system of polynomial equations. Our software, polyTop, leverages numerical algebraic geometry computations from Bertini and Bertini_Real with topological computations in javaPlex to compute the Euler characteristic, genus, Betti numbers, and generators of the fundamental group of a smooth real surface. This poster will highlight several examples that demonstrate this new software.

Michael Ruddy

Title: Signature Polynomials for Algebraic Curves

Abstract: For the action of a group G on the plane, two curves C and C' are G-equivalent if there exists some g in G such that g*C=C'. The group equivalence problem for curves can be stated as: given two curves, decide if they are G-equivalent. The signature method to answer the local group equivalence problem for smooth curves and its application to image science has been studied extensively. However, computing an implicit equation of a signature curve is a challenging problem. In this poster we consider signatures for algebraic curves and show that the degree of the polynomial vanishing on the signature can be predicted without computing the polynomial explicitly. We also present some interesting examples and applications of the degree formula. This is joint work with Dr. Irina Kogan and Dr. Cynthia Vinzant.

Sam Sherman

Title: Statistical estimation of the number of solutions to motion generation problems

Abstract: Large structured polynomial systems, such as those arising in motion generation of linkages in kinematics, tend to have far fewer solutions than traditional upper bounds would suggest. In this talk, we will describe a statistical method for estimating the total number of solutions. The new approach extends previous work on success ratios of parameter homotopies to monodromy loops and adds a trace test stopping criteria for validating that all solutions have been found. Several examples will be presented demonstrating the method including the Watt I six-bar motion generation problems.