Related papers: Approximating Convex Hulls via Range Queries
In this paper, we study the fundamental open question of finding the optimal high-order algorithm for solving smooth convex minimization problems. Arjevani et al. (2019) established the lower bound $\Omega\left(\epsilon^{-2/(3p+1)}\right)$…
We study the question of how to compute a point in the convex hull of an input set $S$ of $n$ points in ${\mathbb R}^d$ in a differentially private manner. This question, which is trivial non-privately, turns out to be quite deep when…
We propose a family of recursive cutting-plane algorithms to solve feasibility problems with constrained memory, which can also be used for first-order convex optimization. Precisely, in order to find a point within a ball of radius…
The maximum volume $j$-simplex problem asks to compute the $j$-dimensional simplex of maximum volume inside the convex hull of a given set of $n$ points in $\mathbb{Q}^d$. We give a deterministic approximation algorithm for this problem…
ROC is usually used to analyze the performance of classifiers in data mining. ROC convex hull (ROCCH) is the least convex major-ant (LCM) of the empirical ROC curve, and covers potential optima for the given set of classifiers. Generally,…
We study the convex hull membership (CHM) problem in the pure exploration setting where one aims to efficiently and accurately determine if a given point lies in the convex hull of means of a finite set of distributions. We give a complete…
We consider the planar dynamic convex hull problem. In the literature, solutions exist supporting the insertion and deletion of points in poly-logarithmic time and various queries on the convex hull of the current set of points in…
Kernel quadrature can exploit RKHS spectral structure and outperform Monte Carlo on smooth integrands, but optimized quadrature weights are generally signed and may be numerically unstable. We study whether spectral acceleration remains…
A landmark result of non-smooth convex optimization is that gradient descent is an optimal algorithm whenever the number of computed gradients is smaller than the dimension $d$. In this paper we study the extension of this result to the…
We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic…
In this paper, we focus on lower bounds and algorithms for some basic geometric problems in the one-pass (insertion only) streaming model. The problems considered are grouped into three categories: (i) Klee's measure (ii) Convex body…
We develop algorithms for the optimization of convex objectives that have H\"older continuous $q$-th derivatives by using a $q$-th order oracle, for any $q \geq 1$. Our algorithms work for general norms under mild conditions, including the…
Consider a set $P \subseteq \Re^d$ of $n$ points, and a convex body $C$ provided via a separation oracle. The task at hand is to decide for each point of $P$ if it is in $C$ using the fewest number of oracle queries. We show that one can…
A set in the Euclidean plane is said to be biconvex if, for some angle $\theta\in[0,\pi/2)$, all its sections along straight lines with inclination angles $\theta$ and $\theta+\pi/2$ are convex sets (i.e, empty sets or segments).…
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…
We analyze the orthogonal greedy algorithm when applied to dictionaries $\mathbb{D}$ whose convex hull has small entropy. We show that if the metric entropy of the convex hull of $\mathbb{D}$ decays at a rate of $O(n^{-\frac{1}{2}-\alpha})$…
Convex hull data structures are fundamental in computational geometry. We study insertion-only data structures, supporting various containment and intersection queries. When $P$ is sorted by $x$- or $y$-coordinate, convex hulls can be…
It is well known that the convex hull of $\{(x,y,xy)\}$, where $(x,y)$ is constrained to lie in a box, is given by the Reformulation-Linearization Technique (RLT) constraints. Belotti {\em et al.\,}(2010) and Miller {\em et al.\,}(2011)…
We derive oracle inequalities for the problems of isotonic and convex regression using the combination of $Q$-aggregation procedure and sparsity pattern aggregation. This improves upon the previous results including the oracle inequalities…
A novel 2-D method for computing the convex hull of a sufficiently dense set of n integer points is introduced. The approach employs a ranking function that avoids sorting the points directly thus reducing the overall time complexity. The…