arXiv subjects

Roan Talbut

Publications and source records attributed to Roan Talbut.

4 recordsLinked to original sources

arXiv2026

On the Uniqueness of Fr\'echet Means for Polytope Norms

Fr\'echet means are a popular type of average for non-Euclidean datasets, defined as those points which minimise the average squared distance to a set of data points. We consider the behaviour of sample Fr\'echet means on normed spaces whose unit ball is a polytope; this setting is rarely covered by existing literature on Fr\'echet means, which focuses on smooth spaces or spaces with bounded curvature. We study the geometry of the set of Fr\'echet means over polytope normed spaces, with a focus on dimension and probabilistic conditions for uniqueness. In particular, we provide a geometric characterisation of the threshold sample size at which Fr\'echet means have a positive probability of being unique, and we prove that this threshold is at most one more than the dimension of our space. We are able to use this geometric characterisation to compute the unique Fr\'echet mean sample threshold in the case of the $\ell_\infty$ and $\ell_1$ norms.

math.PR↗

arXiv2025

Tropical Fermat-Weber Points over Spaces of $M$-Ultrametrics

We extend reconstruction methods for phylogenetic trees to ultrametrics of arbitrary matroids and study the stability of these data analysis methods in the combinatorial spirit of Andreas Dress. In particular, we generalize Atteson's work on the safety radius of phylogenetic reconstruction methods, as well as Gascuel and Steel's work on the stochastic safety radius, to arbitrary matroids. We also show that although the tropical Fermat-Weber points of an $M$-ultrametric sample are generally not contained in the space of $M$-ultrametrics, the intersection between the Fermat-Weber set and the space of $M$-ultrametrics is non-empty.

math.CO↗

arXiv2024

Tropical Gradient Descent

We propose a gradient descent method for solving optimization problems arising in settings of tropical geometry - a variant of algebraic geometry that has attracted growing interest in applications such as computational biology, economics, and computer science. Our approach takes advantage of the polyhedral and combinatorial structures arising in tropical geometry to propose a versatile method for approximating local minima in tropical statistical optimization problems - a rapidly growing body of work in recent years. Theoretical results establish global solvability for 1-sample problems and a convergence rate matching classical gradient descent. Numerical experiments demonstrate the method's superior performance compared to classical gradient descent for tropical optimization problems which exhibit tropical convexity but not classical convexity. We also demonstrate the seamless integration of tropical descent into advanced optimization methods, such as Adam, offering improved overall accuracy.

math.OC↗

arXiv2023

Probability Metrics for Tropical Spaces of Different Dimensions

The problem of comparing probability distributions is at the heart of many tasks in statistics and machine learning. Established comparison methods treat the standard setting that the distributions are supported in the same space. Recently, a new geometric solution has been proposed to address the more challenging problem of comparing measures in Euclidean spaces of differing dimensions. Here, we study the same problem of comparing probability distributions of different dimensions in the tropical setting, which is becoming increasingly relevant in applications involving complex data structures such as phylogenetic trees. Specifically, we construct a Wasserstein distance between measures on different tropical projective tori -- the focal metric spaces in both theory and applications of tropical geometry -- via tropical mappings between probability measures. We prove equivalence of the directionality of the maps, whether mapping from a low dimensional space to a high dimensional space or vice versa. As an important practical implication, our work provides a framework for comparing probability distributions on the spaces of phylogenetic trees with different leaf sets. We demonstrate the computational feasibility of our approach using existing optimisation techniques on both simulated and real data.

math.MG↗