Selection Lemmas for various geometric objects

Selection lemmas are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection lemma type results typically show that there exists a point that is contained in many objects that are induced (spanned) by an underlying point set. In the first selection lemma, we consider the set of all the objects induced (spanned) by a point set $P$. This question has been widely explored for simplices in $\mathbb{R}^d$, with tight bounds in $\mathbb{R}^2$. In our paper, we prove first selection lemma for other classes of geometric objects. We also consider the strong variant of this problem where we add the constraint that the piercing point comes from $P$. We prove an exact result on the strong and the weak variant of the first selection lemma for axis-parallel rectangles, special subclasses of axis-parallel rectangles like quadrants and slabs, disks (for centrally symmetric point sets). We also show non-trivial bounds on the first selection lemma for axis-parallel boxes and hyperspheres in $\mathbb{R}^d$. In the second selection lemma, we consider an arbitrary $m$ sized subset of the set of all objects induced by $P$. We study this problem for axis-parallel rectangles and show that there exists an point in the plane that is contained in $\frac{m^3}{24n^4}$ rectangles. This is an improvement over the previous bound by Smorodinsky and Sharir when $m$ is almost quadratic

Fast Conical Hull Algorithms for Near-separable Non-negative Matrix Factorization

The separability assumption (Donoho & Stodden, 2003; Arora et al., 2012) turns non-negative matrix factorization (NMF) into a tractable problem. Recently, a new class of provably-correct NMF algorithms have emerged under this assumption. In this paper, we reformulate the separable NMF problem as that of finding the extreme rays of the conical hull of a finite set of vectors. From this geometric perspective, we derive new separable NMF algorithms that are highly scalable and empirically noise robust, and have several other favorable properties in relation to existing methods. A parallel implementation of our algorithm demonstrates high scalability on shared- and distributed-memory machines.Comment: 15 pages, 6 figure

A geometric approach to archetypal analysis and non-negative matrix factorization

Archetypal analysis and non-negative matrix factorization (NMF) are staples in a statisticians toolbox for dimension reduction and exploratory data analysis. We describe a geometric approach to both NMF and archetypal analysis by interpreting both problems as finding extreme points of the data cloud. We also develop and analyze an efficient approach to finding extreme points in high dimensions. For modern massive datasets that are too large to fit on a single machine and must be stored in a distributed setting, our approach makes only a small number of passes over the data. In fact, it is possible to obtain the NMF or perform archetypal analysis with just two passes over the data.Comment: 36 pages, 13 figure

Some properties of n-dimensional triangulations

A number of mathematical results relevant to the problem of constructing a triangulation, i.e., a simplicial tessellation, of the convex hull of an arbitrary finite set of points in n-space are described. The principal results achieved are: (1) a set of n+2 points in n-space may be triangulated in at most 2 different ways; (2) the sphere test defined in this report selects a preferred one of these two triangulations; (3) a set of parameters is defined that permits the characterization and enumeration of all sets of n+2 points in n-space that are significantly different from the point of view of their possible triangulation; (4) the local sphere test induces a global sphere test property for a triangulation; and (5) a triangulation satisfying the global sphere property is dual to the n-dimensional Dirichlet tesselation, i.e., it is a Delaunay triangulation

Shock formation for 2D2D quasilinear wave systems featuring multiple speeds: Blowup for the fastest wave, with non-trivial interactions up to the singularity

We prove a stable shock formation result for a large class of systems of quasilinear wave equations in two spatial dimensions. We give a precise description of the dynamics all the way up to the singularity. Our main theorem applies to systems of two wave equations featuring two distinct wave speeds and various quasilinear and semilinear nonlinearities, while the solutions under study are (non-symmetric) perturbations of simple outgoing plane symmetric waves. The two waves are allowed to interact all the way up to the singularity. Our approach is robust and could be used to prove shock formation results for other related systems with many unknowns and multiple speeds, in various solution regimes, and in higher spatial dimensions. However, a fundamental aspect of our framework is that it applies only to solutions in which the "fastest wave" forms a shock while the remaining solution variables do not. Our approach is based on an extended version of the geometric vectorfield method developed by D. Christodoulou in his study of shock formation for scalar wave equations as well as the framework developed in our recent joint work with J. Luk, in which we proved a shock formation result for a quasilinear wave-transport system featuring a single wave operator. A key new difficulty that we encounter is that the geometric vectorfields that we use to commute the equations are, by necessity, adapted to the wave operator of the (shock-forming) fast wave and therefore exhibit very poor commutation properties with the slow wave operator, much worse than their commutation properties with a transport operator. To overcome this difficulty, we rely on a first-order reformulation of the slow wave equation, which, though somewhat limiting in the precision it affords, allows us to avoid uncontrollable commutator terms.Comment: 117 pages, 3 figure

Approximating the least hypervolume contributor: NP-hard in general, but fast in practice

The hypervolume indicator is an increasingly popular set measure to compare the quality of two Pareto sets. The basic ingredient of most hypervolume indicator based optimization algorithms is the calculation of the hypervolume contribution of single solutions regarding a Pareto set. We show that exact calculation of the hypervolume contribution is #P-hard while its approximation is NP-hard. The same holds for the calculation of the minimal contribution. We also prove that it is NP-hard to decide whether a solution has the least hypervolume contribution. Even deciding whether the contribution of a solution is at most (1+\eps) times the minimal contribution is NP-hard. This implies that it is neither possible to efficiently find the least contributing solution (unless $P = NP$) nor to approximate it (unless $NP = BPP$). Nevertheless, in the second part of the paper we present a fast approximation algorithm for this problem. We prove that for arbitrarily given \eps,\delta>0 it calculates a solution with contribution at most (1+\eps) times the minimal contribution with probability at least $(1-\delta)$. Though it cannot run in polynomial time for all instances, it performs extremely fast on various benchmark datasets. The algorithm solves very large problem instances which are intractable for exact algorithms (e.g., 10000 solutions in 100 dimensions) within a few seconds.Comment: 22 pages, to appear in Theoretical Computer Scienc