Poster Session

Numerical Algebraic Geometry meets Differential Signatures

We apply numerical algebraic geometry to the invariant-theoretic problem of detecting symmetries between two plane algebraic curves. We describe an efficient equality test which determines, with ``probability-one'', whether or not two rational maps have the same image up to Zariski closure. The application to invariant theory is based on the construction of suitable signature maps associated to a group acting linearly on the respective curves. We consider two versions of this construction: differential and joint signature maps. In our examples and computational experiments, we focus on the complex Euclidean group, and introduce an algebraic joint signature that we prove determines equivalence of curves under this action. We demonstrate that the test is efficient and use it to empirically compare the sensitivity of differential and joint signatures to noise.

Exact solutions in log-concave maximum likelihood estimation

Shape-constrained density estimation has gained attention in recent years. We focus on the case when the densities on $R^d$ are log-concave. It has been shown that the log of optimal log concave density is piecewise linear and supported on a regular subdivision of the given data sample (Cule, Samworth, Stewart) and that every regular subdivision arises in ML for some set of weights (Robeva, Sturmfels, Uhler). We further the understanding of logconcave MLE by studying its exact solutions and connecting this research to the recent developments in solving of polynomial-exponential systems.

Path Signatures on Lie Groups

Path signatures are powerful nonparametric tools for time series analysis, shown to form a universal and characteristic feature map for Euclidean valued time series data. We lift the theory of path signatures to the setting of Lie group valued time series, adapting these tools for time series with underlying geometric constraints. We prove that this generalized path signature is universal and characteristic. To demonstrate universality, we analyze the human action recognition problem in computer vision, using SO(3) representations for the time series, providing comparable performance to other shallow learning approaches, while offering an easily interpretable feature set. We also provide a two-sample hypothesis test for Lie group-valued random walks to illustrate its characteristic property.

Shape Analysis on Lie Groups (and beyond) with Applications

The shapes of level curves of real polynomials near strict local minima

We consider a real bivariate polynomial function vanishing at the origin and exhibiting a strict local minimum at this point. We work in a neighbourhood of the origin in which the non-zero level curves of this function are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, these level curves may fail to be convex.

The aim of this talk is two-fold. Firstly, to study a combinatorial object measuring this non-convexity; it is a planar rooted tree. And secondly, we want to characterise all possible topological types of these objects. To this end, we construct a family of polynomial functions with non-Morse strict local minima realising a large class of such trees.