Related papers: On the Statistics of Lattice Polytopes
We introduce a new $hp$-adaptive strategy for self-adjoint elliptic boundary value problems that does not rely on using classical a posteriori error estimators. Instead, our approach is based on a generally applicable prediction strategy…
This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The…
Recently, several striking advances have taken place regarding the discrete logarithm problem (DLP) in finite fields of small characteristic, despite progress having remained essentially static for nearly thirty years, with the best known…
Consider the d-dimensional lattice Z^d where each vertex is ``open'' or ``closed'' with probability p or 1-p, respectively. An open vertex v is connected by an edge to the closest open vertex w such that the dth co-ordinates of v and w…
We review models of random geometries based on the dynamical lattice approach. We discuss one dimensional model of simplicial complexes (branched polymers), two dimensional model of dynamical triangulations and four dimensional model of…
Symmetry, which describes invariance, is an eternal concern in mathematics and physics, especially in the investigation of solutions to the partial differential equation (PDE). A PDE's nonlocally related PDE systems provide excellent…
The aim of this paper is to investigate the use of an entropic projection method for the iterative regularization of linear ill-posed problems. We derive a closed form solution for the iterates and analyze their convergence behaviour both…
We introduce the broad subclass of algebraic compressed sensing problems, where structured signals are modeled either explicitly or implicitly via polynomials. This includes, for instance, low-rank matrix and tensor recovery. We employ…
In this paper we investigate the ability of a neural network to approximate algebraic properties associated to lattice simplices. In particular we attempt to predict the distribution of Hilbert basis elements in the fundamental…
A lattice polytope $\mathcal{P}$ is called reflexive if its dual $\mathcal{P}^\vee$ is a lattice polytope. The property that $\mathcal{P}$ is unimodularly equivalent to $\mathcal{P}^\vee$ does not hold in general, and in fact there are few…
In this paper we the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type. We carry out a convergence analysis in the sense of regularization methods and discuss…
The Ehrhart function $L_P(t)$ of a polytope $P$ is usually defined only for integer dilation arguments $t$. By allowing arbitrary real numbers as arguments we may also detect integer points entering (or leaving) the polytope in fractional…
We relate scattering amplitudes in particle physics to maximum likelihood estimation for discrete models in algebraic statistics. The scattering potential plays the role of the log-likelihood function, and its critical points are solutions…
We present new pivot rules for the Simplex method for LPs over 0/1 polytopes. We show that the number of non-degenerate steps taken using these rules is strongly polynomial and even linear in the dimension or in the number of variables. Our…
Consider an infinite graph with nodes initially labeled by independent Bernoulli random variables of parameter p. We address the density classification problem, that is, we want to design a (probabilistic or deterministic) cellular…
The concept of representing a polytope that is associated with some combinatorial optimization problem as a linear projection of a higher-dimensional polyhedron has recently received increasing attention. In this paper (written for the…
We review various combinatorial applications of field theoretical and matrix model approaches to equilibrium statistical physics involving the enumeration of fixed and random lattice model configurations. We show how the structures of the…
We propose a new approach for the problem of relative depth estimation from a single image. Instead of directly regressing over depth scores, we formulate the problem as estimation of a probability distribution over depth and aim to learn…
Single-Index Models are high-dimensional regression problems with planted structure, whereby labels depend on an unknown one-dimensional projection of the input via a generic, non-linear, and potentially non-deterministic transformation. As…
Candelas and Font introduced the notion of a `top' as half of a three dimensional reflexive polytope and noticed that Dynkin diagrams of enhanced gauge groups in string theory can be read off from them. We classify all tops satisfying a…