Related papers: The Polyhedral Geometry of Pivot Rules and Monoton…
Border bases arise as a canonical generalization of Gr\"obner bases. We provide a polyhedral characterization of all order ideals (and hence border bases) that are supported by a zero-dimensional ideal: order ideals that support a border…
A regularized version of Mixture Models is proposed to learn a principal graph from a distribution of $D$-dimensional data points. In the particular case of manifold learning for ridge detection, we assume that the underlying manifold can…
Abstract polytopes are combinatorial structures with distinctive geometric, algebraic, or topological characteristics, that generalize (the face lattice of) traditional polyhedra, polytopes or tessellations. Most research has focused on…
Given a triangulation of a closed, oriented, irreducible, atoroidal 3-manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by…
The existence of a polynomial-time pivot rule for the simplex method is a fundamental open question in optimization. While many super-polynomial lower bounds exist for individual or very restricted classes of pivot rules, there currently is…
We consider the problem of learning the structure of undirected graphical models with bounded treewidth, within the maximum likelihood framework. This is an NP-hard problem and most approaches consider local search techniques. In this…
Circuits play a fundamental role in polyhedral theory and linear programming. For instance, circuits are used as step directions in various augmentation schemes for solving linear programs or to leave degenerate vertices while running the…
We present a new approach to learning the structure and parameters of a Bayesian network based on regularized estimation in an exponential family representation. Here we show that, given a fixed variable order, the optimal structure and…
In the early 1990's, Billera and Sturmfels introduced the monotone path polytope (MPP), a special case of the general theory of fiber polytopes that associates a polytope to a pair $(P,\varphi)$ of a polytope $P$ and linear functional…
Mathematical psychology has a long tradition of modeling probabilistic choice via distribution-free random utility models and associated random preference models. For such models, the predicted choice probabilities often form a bounded and…
Geometric hitting set problems, in which we seek a smallest set of points that collectively hit a given set of ranges, are ubiquitous in computational geometry. Most often, the set is discrete and is given explicitly. We propose new…
We introduce topological notions of polytopes and simplexes, the latter being expected to play in p-adically closed fields the role played by real simplexes in the classical results of triangulation of semi-algebraic sets over real closed…
We survey the Hilbert geometry of convex polytopes. In particular we present two important characterisations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a bi-lipschitz class of…
The edges of the characteristic imset polytope, $\operatorname{CIM}_p$, were recently shown to have strong connections to causal discovery as many algorithms could be interpreted as greedy restricted edge-walks, even though only a strict…
The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but…
We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are…
Recent numerical experiments have demonstrated that the choice of optimization geometry used during training can impact generalization performance when learning expressive nonlinear model classes such as deep neural networks. These…
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…
A principal curve serves as a powerful tool for uncovering underlying structures of data through 1-dimensional smooth and continuous representations. On the basis of optimal transport theories, this paper introduces a novel principal curve…
In many applications involving binary variables, only pairwise dependence measures, such as correlations, are available. However, for multi-way tables involving more than two variables, these quantities do not uniquely determine the joint…