Approximating a given three-dimensional object in order to simplify its handling is a classical topic in computational geometry and related fields. A typical approach is based on incremental approximation algorithms, which start with a small and topologically correct polytope representation (the seed polytope) of a given sample point cloud or input mesh. In addition, a correspondence between th...

An extended formulation of a polytope is a linear description of this polytope using extra variables besides the variables in which the polytope is defined. The interest of extended formulations is due to the fact that many interesting polytopes have extended formulations with a lot fewer inequalities than any linear description in the original space. This motivates the development of methods f...

We present a semi-supervised learning algorithm for learning discrete factor analysis models with arbitrary structure on the latent variables. Our algorithm assumes that every latent variable has an “anchor”, an observed variable with only that latent variable as its parent. Given such anchors, we show that it is possible to consistently recover moments of the latent variables and use these mom...

A sharp upper bound on the probability of a random vector falling outside a polytope, based solely on the first and second moments of its distribution, can be computed efficiently using semidefinite programming. However, this Chebyshev-type bound tends to be overly conservative since it is determined by a discrete worst-case distribution. In this paper we obtain a less pessimistic Gauss-type bo...

A theorem of Delzant states that any symplectic manifold (M,ω) of dimension 2n, equipped with an effective Hamiltonian action of the standard n-torus Tn = Rn/2πZn, is a smooth projective toric variety completely determined (as a Hamiltonian Tn-space) by the image of the moment map φ : M → Rn, a convex polytope P = φ(M) ⊂ Rn. In this paper we show, using symplectic (action-angle) coordinates on ...

The cut polytope of a graph arises in many fields. Although much is known about facets of the cut polytope of the complete graph, very little is known for general graphs. The study of Bell inequalities in quantum information science requires knowledge of the facets of the cut polytope of the complete bipartite graph or, more generally, the complete k-partite graph. Lifting is a central tool to ...

The polytope of pointed pseudo triangulations was described in [RSS01]. This polytope is a combinatorial tool to observe all possible pseudo triangulations of a certain point set. Each point of the polytope refers to one possible pseudo triangulation of the point set. To polytope vertices are connected by an edge if their pseudo triangulations just differ in one edge-flip. Once we have a PPT-po...

Recently a generalization of simple convex polytopes to combinatorial entit ies known as abstract polytopes has been proposed. The graph of an abstract polytope o f d imension d is a regular connected graph of degree d. Given a connected regular graph F o f degree d, it is interesting to find out whether it is the graph of some abstract polytope P. We obtain necessary and sufficient condit ions...

The Sherali-Adams (SA) and Lasserre (LS) approaches are “lift-and-project” methods that generate nested sequences of linear and/or semidefinite relaxations of an arbitrary 0-1 polytope P ⊆ [0, 1]n. Although both procedures are known to terminate with an exact description of P after n steps, there are various open questions associated with characterizing, for particular problem classes, whether ...

These notes accompany a lecture about the topology of symplectic (and other) quotients. The aim is two-fold: first to advertise the ease of computation in the symplectic category; and second to give an account of some new computations for weighted projective spaces. We start with a brief exposition of how orbifolds arise in the symplectic category, and discuss the techniques used to understand ...