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Georg Loho

Publications and source records attributed to Georg Loho.

At least 37 records · Page 2Linked to original sources
arXiv2024

On the Correlation Gap of Matroids

A set function can be extended to the unit cube in various ways; the correlation gap measures the ratio between two natural extensions. This quantity has been identified as the performance guarantee in a range of approximation algorithms and mechanism design settings. It is known that the correlation gap of a monotone submodular function is at least $1-1/e$, and this is tight for simple matroid rank functions. We initiate a fine-grained study of the correlation gap of matroid rank functions. In particular, we present an improved lower bound on the correlation gap as parametrized by the rank and girth of the matroid. We also show that for any matroid, the correlation gap of its weighted matroid rank function is minimized under uniform weights. Such improved lower bounds have direct applications for submodular maximization under matroid constraints, mechanism design, and contention resolution schemes.

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arXiv2024

The Real Tropical Geometry of Neural Networks

We consider a binary classifier defined as the sign of a tropical rational function, that is, as the difference of two convex piecewise linear functions. The parameter space of ReLU neural networks is contained as a semialgebraic set inside the parameter space of tropical rational functions. We initiate the study of two different subdivisions of this parameter space: a subdivision into semialgebraic sets, on which the combinatorial type of the decision boundary is fixed, and a subdivision into a polyhedral fan, capturing the combinatorics of the partitions of the dataset. The sublevel sets of the 0/1-loss function arise as subfans of this classification fan, and we show that the level-sets are not necessarily connected. We describe the classification fan i) geometrically, as normal fan of the activation polytope, and ii) combinatorially through a list of properties of associated bipartite graphs, in analogy to covector axioms of oriented matroids and tropical oriented matroids. Our findings extend and refine the connection between neural networks and tropical geometry by observing structures established in real tropical geometry, such as positive tropicalizations of hypersurfaces and tropical semialgebraic sets.

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arXiv2024

Signed tropical halfspaces and convexity

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 convexity for signed tropical numbers called 'TO-convexity' (formerly 'signed tropical convexity') and the novel notion 'TC-convexity'. We derive several separation results for TO-convexity and TC-convexity. A key ingredient is a thorough understanding of TC-hemispaces - those TC-convex sets whose complement is also TC-convex. Furthermore, we use new insights in the interplay between convexity over Puiseux series and its signed valuation. Remarkably, TC-convexity can be seen as a natural convexity notion for representing oriented matroids as it arises from a generalization of the composition operation of vectors in an oriented matroid. We make this explicit by giving representations of linear spaces over the real tropical hyperfield in terms of TC-convexity.

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arXiv2023

The Worst-Case Complexity of Symmetric Strategy Improvement

Symmetric strategy improvement is an algorithm introduced by Schewe et al. (ICALP 2015) that can be used to solve two-player games on directed graphs such as parity games and mean payoff games. In contrast to the usual well-known strategy improvement algorithm, it iterates over strategies of both players simultaneously. The symmetric version solves the known worst-case examples for strategy improvement quickly, however its worst-case complexity remained open. We present a class of worst-case examples for symmetric strategy improvement on which this symmetric version also takes exponentially many steps. Remarkably, our examples exhibit this behaviour for any choice of improvement rule, which is in contrast to classical strategy improvement where hard instances are usually hand-crafted for a specific improvement rule. We present a generalized version of symmetric strategy iteration depending less rigidly on the interplay of the strategies of both players. However, it turns out it has the same shortcomings.

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arXiv2023

Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes

We prove that the set of functions representable by ReLU neural networks with integer weights strictly increases with the network depth while allowing arbitrary width. More precisely, we show that $\lceil\log_2(n)\rceil$ hidden layers are indeed necessary to compute the maximum of $n$ numbers, matching known upper bounds. Our results are based on the known duality between neural networks and Newton polytopes via tropical geometry. The integrality assumption implies that these Newton polytopes are lattice polytopes. Then, our depth lower bounds follow from a parity argument on the normalized volume of faces of such polytopes.

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arXiv2022

Face posets of tropical polyhedra and monomial ideals

We exhibit several posets arising from commutative algebra, order theory, tropical convexity as potential face posets of tropical polyhedra, and we clarify their inclusion relations. We focus on monomial tropical polyhedra, and deduce how their geometry reflects properties of monomial ideals. Their vertex-facet lattice is homotopy equivalent to a sphere and encodes the Betti numbers of an associated monomial ideal.

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arXiv2022

Generalized permutahedra and positive flag Dressians

We study valuated matroids, their tropical incidence relations, flag matroids and total positivity. This leads to a characterization of permutahedral subdivisions, namely subdivisions of regular permutahedra into generalized permutahedra. Further, we get a characterization of those subdivisions arising from positive valuated flag matroids.

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arXiv2022

Tropical Positivity and Determinantal Varieties

We initiate the study of positive-tropical generators as positive analogues of the concept of tropical bases. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. We focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. Moreover, in the case square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part.

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arXiv2020

Oriented Matroids from Triangulations of Products of Simplices

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 matching fields. The oriented matroid is composed of compatible chirotopes on the cells in a matroid subdivision of the hypersimplex, which might be of independent interest. In particular, we generalize this using the language of matroids over hyperfields, which gives a new approach to construct matroids over hyperfields. A recurring theme in our work is that various tropical constructions can be extended beyond tropicalization with new formulations and proof methods.

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arXiv2020

Patchworking Oriented Matroids

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 variant of Viro's patchworking to derive a topological representation of the oriented matroid directly from the polyhedral structure of the triangulation, hence finding a combinatorial manifestation of patchworking besides tropical algebraic geometry. We achieve this by rephrasing the patchworking procedure as a controlled cell merging process, guided by the structure of tropical oriented matroids. A key insight is a new promising technique to show that the final cell complex is regular.

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arXiv2020

Matching fields and lattice points of simplices

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 matching field from this bijection. This resolves two open questions from Sturmfels and Zelevinsky (1993) on linkage matching fields. For this, we give an explicit construction that associates a bipartite incidence graph of an ordered partition of a common set to each lattice point in a dilated simplex. Given a triangulation of a product of two simplices encoded by a set of spanning trees on a bipartite node set, we similarly prove that the bijection from left to right degree vectors of the trees is enough to recover the triangulation. As additional results, we show a cryptomorphic description of linkage matching fields and characterise the flip graph of a linkage matching field in terms of its prodsimplicial flag complex. Finally, we relate our findings to transversal matroids through the tropical Stiefel map.

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arXiv2020

Monomial tropical cones for multicriteria optimization

We present an algorithm to compute all $n$ nondominated points of a multicriteria discrete optimization problem with $d$ objectives using at most $\mathcal{O}(n^{\lfloor d/2 \rfloor})$ scalarizations. The method is similar to algorithms by Przybylski et al. (2010) and by Klamroth et al. (2015) with the same complexity. As a difference, our method employs a tropical convex hull computation, and it exploits a particular kind of duality which is special for the tropical cones arising. This duality can be seen as a generalization of the Alexander duality of monomial ideals.

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arXiv2019

Tropical Carathéodory with Matroids

Bárány's colorful generalization of Carathéodory's Theorem combines geometrical and combinatorial constraints. Kalai-Meshulam (2005) and Holmsen (2016) generalized Bárány's theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the tropical colorful Carathéodory Theorem of Gaubert-Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. In particular, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth.

math.CO↗
arXiv2019

Signed tropical convexity

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 Puiseux series and hyperoperations. Along the way, we deduce a new Farkas lemma and Fourier-Motzkin elimination without the non-negativity restriction on the variables. This leads to a Minkowski-Weyl theorem for polytopes over the signed tropical numbers.

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arXiv2019

Tropical Ehrhart Theory and Tropical Volume

We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing measures. Our exposition is complemented by a brief study of arising complexity questions.

math.MG↗
arXiv2019

Weighted digraphs and tropical cones

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, arrangements of finitely many tropical halfspaces can be considered via coarsenings of the resulting polyhedral decompositions of $\mathbb{R}^d$. This leads to natural cell decompositions of the tropical projective space $\mathbb{TP}_{\min}^{d-1}$. Our method is to employ a known class of ordinary convex polyhedra naturally associated with weighted digraphs. This way we can relate to and use results from combinatorics and optimization. One outcome is the solution of a conjecture of Develin and Yu (2007).

math.CO↗
arXiv2017

Abstract tropical linear programming

In this paper we develop a combinatorial abstraction of tropical linear programming. This generalizes the search for a feasible point of a system of min-plus-inequalities. It is based on the polyhedral properties of triangulations of the product of two simplices and the combinatorics of the associated set of bipartite graphs with an additional sign information which we call a signed tropical matroid. We demonstrate the connections with the classical simplex method, mean payoff games and scheduling.

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