Related papers: An Output Sensitive Algorithm for Discrete Convex …
We study the impact of nonconvexity on the complexity of nonsmooth optimization, emphasizing objectives such as piecewise linear functions, which may not be weakly convex. We focus on a dimension-independent analysis, slightly modifying a…
The numerical solution of dynamical systems with memory requires the efficient evaluation of Volterra integral operators in an evolutionary manner. After appropriate discretisation, the basic problem can be represented as a matrix-vector…
We give a survey of work on the number of vertices of the convex hull of integer points defined by the system of linear inequalities. Also, we present our improvement of some of these.
The convex hull peeling of a point set consists in taking the convex hull, then removing the extreme points and iterating that procedure until no point remains. The boundary of each hull is called a layer. Following on from [15], we study…
The it Convex Hull Membership(CHM) problem is: Given a point $p$ and a subset $S$ of $n$ points in $\mathbb{R}^m$, is $p \in conv(S)$? CHM is not only a fundamental problem in Linear Programming, Computational Geometry, Machine Learning and…
For a set $\mathcal{D}$ of disks in the plane, its disk graph $G(\mathcal{D})$ is the graph with vertex set $\mathcal{D}$, where two vertices are adjacent if and only if the corresponding disks intersect. Given a set $\mathcal{D}$ of $n$…
Next generation galaxy surveys demand the development of massive ensembles of galaxy mocks to model the observables and their covariances, what is computationally prohibitive using $N$-body simulations. COLA is a novel method designed to…
A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let {\delta}(d, k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational…
Let $A \in \mathbb{Z}^{m \times n}$ be an integer matrix with components bounded by $\Delta$ in absolute value. Cook et al.~(1986) have shown that there exists a universal matrix $B \in \mathbb{Z}^{m' \times n}$ with the following property:…
We study differentially private stochastic optimization in convex and non-convex settings. For the convex case, we focus on the family of non-smooth generalized linear losses (GLLs). Our algorithm for the $\ell_2$ setting achieves optimal…
Depth measures quantify central tendency in the analysis of statistical and geometric data. Selecting a depth measure that is simple and efficiently computable is often important, e.g., when calculating depth for multiple query points or…
We study the convergence behavior of the stochastic heavy-ball method with a small stepsize. Under a change of time scale, we approximate the discrete method by a stochastic differential equation that models small random perturbations of a…
The convex hull peeling of a point set is obtained by taking the convex hull of the set and repeating iteratively the operation on the interior points until no point remains. The boundary of each hull is called a layer. We study the number…
We consider bounded integer knapsacks where the weights and variable upper bounds together form a superincreasing sequence. The elements of this superincreasing knapsack are exactly those vectors that are lexicographically smaller than the…
We show that the volume of a convex body in $\mathbb{R}^{n}$ in the general membership oracle model can be computed to within relative error $\varepsilon$ using $\widetilde{O}(n^{3.5}\psi^{2} + n^3/\varepsilon^{2})$ oracle queries, where…
Given a set $S$ of $n$ points in the Euclidean plane, the two-center problem is to find two congruent disks of smallest radius whose union covers all points of $S$. Previously, Eppstein [SODA'97] gave a randomized algorithm of $O(n\log^2n)$…
Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let $\Co(A)$ be the convex hull and $\diam(A)$ the diameter of…
Quickhull is an algorithm for computing the convex hull of points in a plane that performs well in practice, but has poor complexity on adversarial input. In this paper we show the same holds for the numerical stability of Quickhull.
We consider the convex hull of the perturbed point process comprised of $n$ i.i.d. points, each distributed as the sum of a uniform point on the unit sphere $\S^{d-1}$ and a uniform point in the $d$-dimensional ball centered at the origin…
Sampling logconcave functions arising in statistics and machine learning has been a subject of intensive study. Recent developments include analyses for Langevin dynamics and Hamiltonian Monte Carlo (HMC). While both approaches have…