Polytopes and Machine Learning

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...

Ubiquitous geometric objects can be precisely and efficiently represented as polyhedra. The transformation of a polyhedron into a vector, known as polyhedra representation learning, is crucial for manipulating these shapes with mathematical and statistical tools for tasks like classification, clustering, and generation. Recent years have witnessed significant strides in this domain, yet most efforts focus on the vertex sequence of a polyhedron, neglecting the complex surface ...

In 1980, V. I. Arnold studied the classification problem for convex lattice polygons of a given area. Since then, this problem and its analogues have been studied by many authors, including B\'ar\'any, Lagarias, Pach, Santos, Ziegler and Zong. Despite extensive study, the structure of the representative sets in the classifications remains unclear, indicating a need for refined classification methods. In this paper, we propose a novel classification framework based on affine e...

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...

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.

Polytopes are one of the most primitive concepts underlying geometry. Discovery and study of polytopes with complex structures provides a means of advancing scientific knowledge. Construction of polytopes with specific extremal structure is very difficult and time-consuming. Having an automated tool for the generation of such extremal examples is therefore of great value. We present an Artificial Intelligence system capable of generating novel polytopes with very high complex...

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...

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...

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....

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...