ID: 2109.09602

Polytopes and Machine Learning

September 16, 2021

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Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations

May 2, 2008

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Jean Gallier
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Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. I have included a rather th...

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Classification of smooth lattice polytopes with few lattice points

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Benjamin Lorenz
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After giving a short introduction on smooth lattice polytopes, I will present a proof for the finiteness of smooth lattice polytopes with few lattice points. The argument is then turned into an algorithm for the classification of smooth lattice polytopes in fixed dimension with an upper bound on the number of lattice points. Additionally I have implemented this algorithm for dimension two and three and used it, together with a classification of smooth minimal fans by Tadao Od...

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Problems on Polytopes, Their Groups, and Realizations

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Egon Schulte, Asia Ivic Weiss
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The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2005.

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How Could Polyhedral Theory Harness Deep Learning?

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Thiago Serra, Christian Tjandraatmadja, Srikumar Ramalingam
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The holy grail of deep learning is to come up with an automatic method to design optimal architectures for different applications. In other words, how can we effectively dimension and organize neurons along the network layers based on the computational resources, input size, and amount of training data? We outline promising research directions based on polyhedral theory and mixed-integer representability that may offer an analytical approach to this question, in contrast to t...

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Geometric Disentanglement by Random Convex Polytopes

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Michael Joswig, Marek Kaluba, Lukas Ruff
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We propose a new geometric method for measuring the quality of representations obtained from deep learning. Our approach, called Random Polytope Descriptor, provides an efficient description of data points based on the construction of random convex polytopes. We demonstrate the use of our technique by qualitatively comparing the behavior of classic and regularized autoencoders. This reveals that applying regularization to autoencoder networks may decrease the out-of-distribut...

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Likelihood Geometry of Reflexive Polytopes

November 22, 2023

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Carlos Améndola, Janike Oldekop
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We study the problem of maximum likelihood (ML) estimation for statistical models defined by reflexive polytopes. Our focus is on the maximum likelihood degree of these models as an algebraic measure of complexity of the corresponding optimization problem. We compute the ML degrees of all 4319 classes of three-dimensional reflexive polytopes, and observe some surprising behavior in terms of the presence of gaps between ML degrees and degrees of the associated toric varieties....

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A Fast and Practical Method to Estimate Volumes of Convex Polytopes

December 31, 2013

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Cunjing Ge, Feifei Ma, Jian Zhang
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The volume is an important attribute of a convex body. In general, it is quite difficult to calculate the exact volume. But in many cases, it suffices to have an approximate value. Volume estimation methods for convex bodies have been extensively studied in theory, however, there is still a lack of practical implementations of such methods. In this paper, we present an efficient method which is based on the Multiphase Monte-Carlo algorithm to estimate volumes of convex polyto...

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Realizability of Polytopes as a Low Rank Matrix Completion Problem

December 16, 2010

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Michael Gene Dobbins
Combinatorics

This article gives necessary and sufficient conditions for a relation to be the containment relation between the facets and vertices of a polytope. Also given here, are a set of matrices parameterizing the linear moduli space and another set parameterizing the projective moduli space of a combinatorial polytope.

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Learning polytopes with fixed facet directions

January 10, 2022

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Maria Dostert, Katharina Jochemko
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We consider the task of reconstructing polytopes with fixed facet directions from finitely many support function evaluations. We show that for a fixed simplicial normal fan the least-squares estimate is given by a convex quadratic program. We study the geometry of the solution set and give a combinatorial characterization for the uniqueness of the reconstruction in this case. We provide an algorithm that, under mild assumptions, converges to the unknown input shape as the num...

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Polymake and Lattice Polytopes

February 17, 2009

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Michael Joswig, Benjamin Müller, Andreas Paffenholz
Combinatorics
Algebraic Geometry

The polymake software system deals with convex polytopes and related objects from geometric combinatorics. This note reports on a new implementation of a subclass for lattice polytopes. The features displayed are enabled by recent changes to the polymake core, which will be discussed briefly.

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