Related papers: Counting faces of randomly-projected polytopes whe…
Let $\mathcal Z_d$ be the zero cell of a $d$-dimensional, isotropic and stationary Poisson hyperplane tessellation. We study the asymptotic behavior of the expected number of $k$-dimensional faces of $\mathcal Z_d$, as $d\to\infty$. For…
Let K be a compact convex body in Rd, let Kn be the convex hull of n points chosen uniformly and independently in K, and let fi(Kn) denote the number of i-dimensional faces of Kn. We show that for planar convex sets, E(f0(Kn)) is increasing…
There are (at least) two reasons to study random polytopes. The first is to understand the combinatorics and geometry of random polytopes especially as compared to other classes of polytopes, and the second is to analyze average-case…
Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of several convex polytopes exactly equals the normalized…
We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are…
We consider linear regression problems with a varying number of random projections, where we provably exhibit a double descent curve for a fixed prediction problem, with a high-dimensional analysis based on random matrix theory. We first…
We study the expected volume of random polytopes generated by taking the convex hull of independent identically distributed points from a given distribution. We show that for log-concave distributions supported on convex bodies, we need at…
Three-dimensional reconstruction is a fundamental problem in robotics perception. We examine the problem of active view selection to perform 3D Gaussian Splatting reconstructions with as few input images as possible. Although 3D Gaussian…
The large sample theory of estimators for density modes is well understood. In this paper we consider density ridges, which are a higher-dimensional extension of modes. Modes correspond to zero-dimensional, local high-density regions in…
Face occlusions, covering either the majority or discriminative parts of the face, can break facial perception and produce a drastic loss of information. Biometric systems such as recent deep face recognition models are not immune to…
Random projection (RP) is a classical technique for reducing storage and computational costs. We analyze RP-based approximations of convex programs, in which the original optimization problem is approximated by the solution of a…
In this paper, we propose an effective face completion algorithm using a deep generative model. Different from well-studied background completion, the face completion task is more challenging as it often requires to generate semantically…
Not only is the geometry of rock fragments often well approximated by ideal convex polyhedra having few faces and vertices, but these numbers carry vital geophysical information on the fragmentation process. Despite their significance, the…
We present a simple and efficient acceleration technique for an arbitrary method for computing the Euclidean projection of a point onto a convex polytope, defined as the convex hull of a finite number of points, in the case when the number…
We study extremal properties of spherical random polytopes, the convex hull of random points chosen from the unit Euclidean sphere in $\mathbb{R}^n$. The extremal properties of interest are the expected values of the maximum and minimum…
Estimating the number of vertices of a two dimensional projection, called a shadow, of a polytope is a fundamental tool for understanding the performance of the shadow simplex method for linear programming among other applications. We prove…
Convex hulls are a fundamental geometric tool used in a number of algorithms. As a side-effect of exhaustive tests for an algorithm for which a convex hull computation was the first step, interesting experimental results were found and are…
According to Euler's relation any polytope P has as many faces of even dimension as it has faces of odd dimension. As a generalization of this fact one can compare the number of faces whose dimension is congruent to i modulo m with the…
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.…
The random beta polytope is defined as the convex hull of $n$ independent random points with the density proportional to $(1-\|x\|^2)^\beta$ on the $d$-dimensional unit ball, where $\beta>-1$ is a parameter. Similarly, the random beta'…