Related papers: A Simple Convex Layers Algorithm

In many cases the convexity of the image of a linear map with range is $R^n$ is automatic because of the facial structure of the domain of the map. We develop a four step procedure for proving this kind of ``automatic convexity''. To make…

Many fundamental problems in computational geometry admit no algorithm running in $o(n \log n)$ time for $n$ planar input points, via classical reductions from sorting. Prominent examples include the computation of convex hulls, quadtrees,…

This document introduces a strategy to solve linear optimization problems. The strategy is based on the bounding condition each constraint produces on each one of the problem's dimension. The solution of a linear optimization problem is…

Deep convolutional neural networks achieve remarkable visual recognition performance, at the cost of high computational complexity. In this paper, we have a new design of efficient convolutional layers based on three schemes. The 3D…

In an ordinary feature selection procedure, a set of important features is obtained by solving an optimization problem such as the Lasso regression problem, and we expect that the obtained features explain the data well. In this study,…

Let $P$ be a $d$-dimensional $n$-point set. A partition $T$ of $P$ is called a Tverberg partition if the convex hulls of all sets in $T$ intersect in at least one point. We say $T$ is $t$-tolerant if it remains a Tverberg partition after…

We address the problem of minimizing a convex function over the space of large matrices with low rank. While this optimization problem is hard in general, we propose an efficient greedy algorithm and derive its formal approximation…

A lot of effort has been invested into characterizing the convergence rates of gradient based algorithms for non-linear convex optimization. Recently, motivated by large datasets and problems in machine learning, the interest has shifted…

A well-known result in the study of convex polyhedra, due to Minkowski, is that a convex polyhedron is uniquely determined (up to translation) by the directions and areas of its faces. The theorem guarantees existence of the polyhedron…

Convex nonsmooth optimization problems, whose solutions live in very high dimensional spaces, have become ubiquitous. To solve them, the class of first-order algorithms known as proximal splitting algorithms is particularly adequate: they…

We study the online convex covering problem and online convex packing problem. The (offline) convex covering problem is modeled by the following convex program: $\min_{x \in R_+^n} f(x) \ \text{s.t}\ A x \ge 1$, where $f : R_+^n \mapsto…

Given a convex region in the plane, and a sweep-line as a tool, what is best way to reduce the region to a single point by a sequence of sweeps? The problem of sweeping points by orthogonal sweeps was first studied in [2]. Here we consider…

We establish a bound of $O(n^2k^{1+\eps})$, for any $\eps>0$, on the combinatorial complexity of the set $\T$ of line transversals of a collection $\P$ of $k$ convex polyhedra in $\reals^3$ with a total of $n$ facets, and present a…

We consider the task of decentralized minimization of the sum of smooth strongly convex functions stored across the nodes of a network. For this problem, lower bounds on the number of gradient computations and the number of communication…

We consider rather a general class of multi-level optimization problems, where a convex objective function is to be minimized subject to constraints of optimality of nested convex optimization problems. As a special case, we consider a…

We propose new algorithms for topic modeling when the number of topics is unknown. Our approach relies on an analysis of the concentration of mass and angular geometry of the topic simplex, a convex polytope constructed by taking the convex…

We study the problem of minimum enclosing rectangle with outliers, which asks to find, for a given set of $n$ planar points, a rectangle with minimum area that encloses at least $(n-t)$ points. The uncovered points are regarded as outliers.…

This paper proposes a novel and simple algorithm of facet enumeration for convex polytopes. The complexity of the algorithm is discussed. The algorithm is implemented in Matlab. Some simple polytopes with known H-representations and…

We introduce a general mathematical framework for distributed algorithms, and a monotonicity property frequently satisfied in application. These properties are leveraged to provide finite-time guarantees for converging algorithms, suited…

We consider the following problem in computational geometry: given, in the d-dimensional real space, a set of points marked as positive and a set of points marked as negative, such that the convex hull of the positive set does not intersect…