33 citations · 62 across the 20 of their papers we have counts for
12 papers · 1 filter
Two Linear Programs are Better than One
Alexander E. Black, Christoph Hertrich, Stefan Kober +1
The extension complexity of a polytope is the minimum number of facets of a polytope that projects onto it. Motivated by the complexity of neural network…
Arithmetic Circuits and Neural Networks for Regular Matroids
Christoph Hertrich, Stefan Kober, Georg Loho
We prove that there exist uniform -circuits of size to compute the basis generating polynomial of regular matroids on elements. By tropicalization, this…
Neural Networks and (Virtual) Extended Formulations
Christoph Hertrich, Georg Loho
Neural networks with piecewise linear activation functions, such as rectified linear units (ReLU) or maxout, are among the most fundamental models in modern machine learning. We ma…
Signed tropical halfspaces and convexity
Georg Loho, Mateusz Skomra
We extend the fundamentals for tropical convexity beyond the tropically positive orthant expanding the theory developed by Loho and Végh (ITCS 2020). We study two notions of convex…
Generalized permutahedra and positive flag Dressians
Michael Joswig, Georg Loho, Dante Luber +1
We study valuated matroids, their tropical incidence relations, flag matroids and total positivity. This leads to a characterization of permutahedral subdivisions, namely subdivisi…
On complete classes of valuated matroids
Edin Husić, Georg Loho, Ben Smith +1
We characterize a rich class of valuated matroids, called R-minor valuated matroids that includes the indicator functions of matroids, and is closed under operations such as taking…
Patchworking Oriented Matroids
Marcel Celaya, Georg Loho, Chi Ho Yuen
In a previous work, we gave a construction of (not necessarily realizable) oriented matroids from a triangulation of a product of two simplices. In this follow-up paper, we use a v…
Oriented Matroids from Triangulations of Products of Simplices
Marcel Celaya, Georg Loho, Chi Ho Yuen
We introduce a construction of oriented matroids from a triangulation of a product of two simplices. For this, we use the structure of such a triangulation in terms of polyhedral m…
Tropical Carathéodory with Matroids
Georg Loho, Raman Sanyal
Bárány's colorful generalization of Carathéodory's Theorem combines geometrical and combinatorial constraints. Kalai-Meshulam (2005) and Holmsen (2016) generalized Bárány's theorem…
Face posets of tropical polyhedra and monomial ideals
Georg Loho, Ben Smith
We exhibit several posets arising from commutative algebra, order theory, tropical convexity as potential face posets of tropical polyhedra, and we clarify their inclusion relation…
Matching fields and lattice points of simplices
Georg Loho, Ben Smith
We show that the Chow covectors of a linkage matching field define a bijection between certain degree vectors and lattice points, and we demonstrate how one can recover the linkage…
Weighted digraphs and tropical cones
Michael Joswig, Georg Loho
This paper is about the combinatorics of finite point configurations in the tropical projective space or, dually, of arrangements of finitely many tropical hyperplanes. Moreover, a…