Related papers: On the Geometry of Maximum Entropy Problems
Random tilings are interesting as idealizations of atomistic models of quasicrystals and for their connection to problems in combinatorics and algorithms. Of particular interest is the tiling entropy density, which measures the relation of…
We consider the problem of embedding the nodes of a hypergraph into Euclidean space under the assumption that the interactions arose through closeness to unknown hyperedge centres. In this way, we tackle the inverse problem associated with…
The classical problem of moments is addressed by the maximum entropy approach for one-dimensional discrete distributions. The numerical technique of adaptive support approximation is proposed to reconstruct the distributions in the region…
Realistic models of biological processes typically involve interacting components on multiple scales, driven by changing environment and inherent stochasticity. Such models are often analytically and numerically intractable. We revisit a…
The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class of optimization problems is strongly utilized in engineering, biology and finance. In this paper, a stochastic…
We establish that the optimal bound for the size of the smallest integral solution of the Oppenheim Diophantine approximation problem $\abs{Q(x)-\xi}< \epsilon$ for a generic ternary form $Q$ is $\abs{x}\ll \epsilon^{-1}$. We also establish…
Entropy is a measure of heterogeneity widely used in applied sciences, often when data are collected over space. Recently, a number of approaches has been proposed to include spatial information in entropy. The aim of entropy is to…
Entropy, under a variety of names, has long been used as a measure of diversity in ecology, as well as in genetics, economics and other fields. There is a spectrum of viewpoints on diversity, indexed by a real parameter q giving greater or…
We adapt ideas and concepts developed in optimal transport (and its martingale variant) to give a geometric description of optimal stopping times of Brownian motion subject to the constraint that the distribution of the stopping time is a…
New algorithms are devised for finding the maxima of multidimensional point samples, one of the very first problems studied in computational geometry. The algorithms are very simple and easily coded and modified for practical needs. The…
The standard implementation of the Maximum Entropy Method (MEM) follows Bryan and deploys a Singular Value Decomposition (SVD) to limit the dimensionality of the underlying solution space apriori. Here we present arguments based on the…
We consider eigenvalue problems for general elliptic operators of arbitrary order subject to homogeneous boundary conditions on open subsets of the euclidean N-dimensional space. We prove stability results for the dependence of the…
We introduce a novel Entropy-driven Monte Carlo (EdMC) strategy to efficiently sample solutions of random Constraint Satisfaction Problems (CSPs). First, we extend a recent result that, using a large-deviation analysis, shows that the…
It is shown that for any ensemble, whether classical or quantum, continuous or discrete, there is only one measure of the "volume" of the ensemble that is compatible with several basic geometric postulates. This volume measure is thus a…
The maximum entropy principle advocates to evaluate events' probabilities using a distribution that maximizes entropy among those that satisfy certain expectations' constraints. Such principle can be generalized for arbitrary decision…
We introduce a new optimization problem regarding embeddings of a graph into a Euclidean space and discuss its relation to the two, mutually dual, optimizations problems introduced by Goering-Helmberg-Wappler. We prove that the Laplace…
We introduce a simple, efficient and precise polynomial heuristic for a key NP complete problem, minimum vertex cover. Our method is iterative and operates in probability space. Once a stable probability solution is found we find the true…
Maximum entropy distributions with discrete support in $m$ dimensions arise in machine learning, statistics, information theory, and theoretical computer science. While structural and computational properties of max-entropy distributions…
Considering the isentropic Euler equations of compressible fluid dynamics with geometric effects included, we establish the existence of entropy solutions for a large class of initial data. We cover fluid flows in a nozzle or in spherical…
We produce a probabilistic space from logic, both classical and quantum, which is in addition partially ordered in such a way that entropy is monotone. In particular do we establish the following equation: Quantitative Probability = Logic +…