Related papers: Tropical balls, geodesics and honeycomb
We establish a new notion of tropical convexity for signed tropical numbers. We provide several equivalent descriptions involving balance relations and intersections of open halfspaces as well as the image of a union of polytopes over…
We revisit the Universal Approximation Theorem(UAT) through the lens of the tropical geometry of neural networks and introduce a constructive, geometry-aware initialization for sigmoidal multi-layer perceptrons (MLPs). Tropical geometry…
Complementing our previous results, we give a classification of all isometries (not necessarily surjective) of the metric space consisting of ball-bodies, endowed with the Hausdorff metric. "Ball bodies" are convex bodies which are…
These condensed notes treat some basic notions in Tropical Geometry (varieties, cycles, modifications, equivalence). These topics are to be extended, illustrated and included to the upcoming book project…
Tropical mathematics is used to establish a correspondence between certain microscopic and macroscopic objects in statistical models. Tropical algebra gives a common framework for macrosystems (subsets) and their elementary constituents…
Tropical counting tools are useful for many enumerative questions. We count tropical multinodal surfaces using floor plans, looking at the case when two nodes are tropically close together, i.e., unseparated. We generalize tropical floor…
We prove general Cramer type theorems for linear systems over various extensions of the tropical semiring, in which tropical numbers are enriched with an information of multiplicity, sign, or argument. We obtain existence or uniqueness…
Monotone path polytopes arise as a special case of the construction of fiber polytopes, introduced by Billera and Sturmfels. A simple example is provided by the permutahedron, which is a monotone path polytope of the standard unit cube. The…
Elements of the tropical vertex group, introduced by Kontsevich and Soibelman, are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are…
A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological…
Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials…
In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…
The term "tropical convexity" was coined by Develin and Sturmfels who published a landmark paper with that title in 2004. However, the topic has much older roots and is deeply connected to linear and combinatorial optimization and other…
This paper is devoted to the bounding and computation of the dimension of deformation spaces of tropical curves and hypersurfaces. This characteristic is interesting in light of the fact that it often coincides with the dimension of…
This paper presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are…
HCMU surfaces are compact Riemann surfaces equipped with the Calabi extremal K\"{a}hler metric and a finite number of singularities. By using both the classical football decomposition introduced by Chen-Chen-Wu and the description of the…
Develin and Sturmfels showed that regular triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ can be thought as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with…
We consider the metric space of all toric K\"ahler metrics on a compact toric manifold; when "looking at it from infinity" (following Gromov), we obtain the tangent cone at infinity, which is parametrized by equivalence classes of complete…
The notion of a tropical vector bundle on a toric variety was recently introduced by Khan-Maclagan and Kaveh-Manon. In this paper, we study the Euler characteristic and rank of global sections for tropical vector bundles. We associate a…
In this paper we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus of this paper is the study…