Related papers: A New and Faster Representation for Counting Integ…

This paper presents in detail the originally developed Quadratic Point Estimate Method (QPEM), aimed at efficiently and accurately computing the first four output moments of probabilistic distributions, using 2n^2+1 sample (or sigma)…

In this article we obtain new irrationality measures for values of functions which belong to a certain class of hypergeometric functions including shifted logarithmic functions, binomial functions and shifted exponential functions. We…

We investigate the folding problem that asks if a polygon P can be folded to a polyhedron Q for given P and Q. Recently, an efficient algorithm for this problem has been developed when Q is a box. We extend this idea to regular polyhedra,…

This paper primarily focuses on computing the Euclidean projection of a vector onto the $\ell_{p}$ ball in which $p\in(0,1)$. Such a problem emerges as the core building block in statistical machine learning and signal processing tasks…

$\renewcommand{\Re}{\mathbb{R}}\newcommand{\eps}{{\varepsilon}}\newcommand{\poly}{\mathrm{poly}} $In this paper, we study the problem of $L_1$-fitting a shape to a set of $n$ points in $\Re^d$ (where $d$ is a fixed constant), where the…

Partial permutohedra are lattice polytopes which were recently introduced and studied by Heuer and Striker. For positive integers $m$ and $n$, the partial permutohedron $\mathcal{P}(m,n)$ is the convex hull of all vectors in…

We investigate properties of Ehrhart polynomials for matroid polytopes, independence matroid polytopes, and polymatroids. In the first half of the paper we prove that for fixed rank their Ehrhart polynomials are computable in polynomial…

This dissertation presents new results on three different themes all related to matroid polytopes. First we investigate properties of Ehrhart polynomials of matroid polytopes, independence matroid polytopes, and polymatroids. We prove that…

A lattice polytope is "free" (or "empty") if its vertices are the only lattice points it contains. In the context of valuation theory, Klain (1999) proposed to study the functions $\alpha_i(P;n)$ that count the number of free polytopes in…

We describe a method for computing the highest degree coefficients of a weighted Ehrhart quasi-polynomial for a rational simple polytope.

We design new polynomials for representing threshold functions in three different regimes: probabilistic polynomials of low degree, which need far less randomness than previous constructions, polynomial threshold functions (PTFs) with…

We give a {\em deterministic} algorithm for approximately computing the fraction of Boolean assignments that satisfy a degree-$2$ polynomial threshold function. Given a degree-2 input polynomial $p(x_1,\dots,x_n)$ and a parameter $\eps >…

This paper addresses the problem of finding a B-term wavelet representation of a given discrete function $f \in \real^n$ whose distance from f is minimized. The problem is well understood when we seek to minimize the Euclidean distance…

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…

For a given sequence $\mathbf{\alpha} = [\alpha_1,\alpha_2,\dots,\alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\mathbf{\alpha})(t)$ that counts the nonnegative integer solutions of the equation…

We consider the problem of computing sample points in each connected component of a semi-algebraic set defined by the non-vanishing or the positivity of an n-variate polynomial of degree d, with rational coefficients of bit size bounded by…

Important problems across multiple disciplines involve computations on the semiring $(\times, \max)$ (or its equivalents, the negated version $(\times, \min)$), the log-transformed version $(+, \max)$, or the negated log-transformed version…

There are many space subdivision and space partitioning techniques used in many algorithms to speed up computations. They mostly rely on orthogonal space subdivision, resp. using hierarchical data structures, e.g. BSP trees, quadtrees,…

Let P be a convex polytope not simple in general. In the focus of this paper lies a simplicial complex K_P which carries complete information about the combinatorial type of P. In the case when P is simple, K_P is the same as dP*, where P*…

The recently developed matrix based Renyi's entropy enables measurement of information in data simply using the eigenspectrum of symmetric positive semi definite (PSD) matrices in reproducing kernel Hilbert space, without estimation of the…