Given a set @math of @math points in the plane, we study the computation of the probability distribution function of both the area and perimeter of the convex hull of a random subset @math of @math . The random subset @math is formed by drawing each point @math of @math independently with a given rational probability @math . For both measures of the convex hull, we show that it is #P-hard to compute the probability that the measure is at least a given bound @math . For @math , we provide an algorithm that runs in @math time and returns a value that is between the probability that the area is at least @math , and the probability that the area is at least @math . For the perimeter, we show a similar algorithm running in @math time. Finally, given @math and for any measure, we show an @math -time Monte Carlo algorithm that returns a value that, with probability of success at least @math , differs at most @math from the probability that the measure is at least @math .
With respect to the convex hull of stochastic points, in the same model that we consider (called unipoint model @cite_0 ), @cite_17 investigated the most likely convex hull of stochastic points, which is the convex hull that appears with the most probability. They proved that such a convex hull can be computed in @math time in the plane, and its computation is NP-hard in higher dimensions.
@cite_0 @cite_17
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Consider a set of points in d dimensions where the existence or the location of each point is determined by a probability distribution. The convex hull of this set is a random variable distributed over exponentially many choices. We are interested in finding the most likely convex hull, namely, the one with the maximum probability of occurrence. We investigate this problem under two natural models of uncertainty: the point (also called the tuple) model where each point (site) has a fixed position s i but only exists with some probability π i , for 0 < π i ≤ 1, and the multipoint model where each point has multiple possible locations or it may not appear at all. We show that the most likely hull under the point model can be computed in O(n 3) time for n points in d = 2 dimensions, but it is NP–hard for d ≥ 3 dimensions. On the other hand, we show that the problem is NP–hard under the multipoint model even for d = 2 dimensions. We also present hardness results for approximating the probability of the most likely hull. While we focus on the most likely hull for concreteness, our results hold for other natural definitions of a probabilistic hull. We study the convex-hull problem in a probabilistic setting, motivated by the need to handle data uncertainty inherent in many applications, including sensor databases, location-based services and computer vision. In our framework, the uncertainty of each input point is described by a probability distribution over a finite number of possible locations including a null location to account for non-existence of the point. Our results include both exact and approximation algorithms for computing the probability of a query point lying inside the convex hull of the input, time–space tradeoffs for the membership queries, a connection between Tukey depth and membership queries, as well as a new notion of ( )-hull that may be a useful representation of uncertain hulls. We study the convex-hull problem in a probabilistic setting, motivated by the need to handle data uncertainty inherent in many applications, including sensor databases, location-based services and computer vision. In our framework, the uncertainty of each input site is described by a probability distribution over a finite number of possible locations including a location to account for non-existence of the point. Our results include both exact and approximation algorithms for computing the probability of a query point lying inside the convex hull of the input, time-space tradeoffs for the membership queries, a connection between Tukey depth and membership queries, as well as a new notion of @math -hull that may be a useful representation of uncertain hulls. In this paper we present several results on the expected complexity of a convex hull of @math points chosen uniformly and independently from a convex shape. (i) We show that the expected number of vertices of the convex hull of @math points, chosen uniformly and independently from a disk is @math , and @math for the case a convex polygon with @math sides. Those results are well known (see rs-udkhv-63,r-slcdn-70,ps-cgi-85 ), but we believe that the elementary proof given here are simpler and more intuitive. (ii) Let @math be a set of directions in the plane, we define a generalized notion of convexity induced by @math , which extends both rectilinear convexity and standard convexity. We prove that the expected complexity of the @math -convex hull of a set of @math points, chosen uniformly and independently from a disk, is @math , where @math is the largest angle between two consecutive vectors in @math . This result extends the known bounds for the cases of rectilinear and standard convexity. (iii) Let @math be an axis parallel hypercube in @math . We prove that the expected number of points on the boundary of the quadrant hull of a set @math of @math points, chosen uniformly and independently from @math is @math . Quadrant hull of a set of points is an extension of rectilinear convexity to higher dimensions. In particular, this number is larger than the number of maxima in @math , and is also larger than the number of points of @math that are vertices of the convex hull of @math . Those bounds are known bkst-anmsv-78 , but we believe the new proof is simpler. In this paper we characterize the convex hull of feasible points for a disjunctive program, a class of problems which subsumes pure and mixed integer programs and many other nonconvex programming problems. Two representations are given for the convex hull of feasible points, each of which provides linear programming equivalents of the disjunctive program. The first one involves a number of new variables proportional to the number of terms in the disjunctive normal form of the logical constraints; the second one involves only the original variables and the facets of the convex hull. Among other results, we give necessary and sufficient conditions for an inequality to define a facet of the convex hull of feasible points. For the class of disjunctive programs that we call facial, we establish a property which makes it possible to obtain the convex hull of points satisfying n disjunctions, in a sequence of n steps, where each step generates the convex hull of points satisfying one disjunction only. We describe an @math -time algorithm for computing the exact probability that two @math -dimensional probabilistic point sets are linearly separable, for any fixed @math . A probabilistic point in @math -space is a normal point, but with an associated probability of existence; the existence probabilities of all points are independent. We also show that the @math -dimensional separability problem is equivalent to a @math -dimensional convex hull membership problem, which asks for the probability that a query point lies inside the convex hull of @math probabilistic points. Using this reduction, we improve the current best bound for the convex hull membership problem by a factor of @math . In addition, our algorithms can handle input degeneracies in which more than @math points may lie on a @math -dimensional subspace, thus resolving an open problem in 2013. Finally, we prove lower bounds for the separability problem via a reduction from the @math -SUM problem, which show in particular that our @math algorithms for @math -dimensional separability and @math -dimensional convex hull membership are nearly optimal. In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint multipoint uncertainty models. Let S=S_R U S_B be a given set of stochastic bichromatic points, and define n = min |S_R|, |S_B| and N = max |S_R|, |S_B| . We show that the separable-probability (SP) of S can be computed in O(nN^ d-1 ) time for d >= 3 and O(min nN log N, N^2 ) time for d=2, while the expected separation-margin (ESM) of S can be computed in O(nN^d) time for d >= 2. In addition, we give an Omega(nN^ d-1 ) witness-based lower bound for computing SP, which implies the optimality of our algorithm among all those in this category. Also, a hardness result for computing ESM is given to show the difficulty of further improving our algorithm. As an extension, we generalize the same problems from points to general geometric objects, i.e., polytopes and or balls, and extend our algorithms to solve the generalized SP and ESM problems in O(nN^d) and O(nN^ d+1 ) time, respectively. Finally, we present some applications of our algorithms to stochastic convex-hull related problems. The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it used less memory. computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating-point arithmetic, this assumption can lead to serous errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions. We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O(log3n) expected amortized time, deletions take O(log6n) expected amortized time, and extreme-point queries take O(log2n) worst-case time. This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous O(ne)-time method by Agarwal and Matousek a decade ago. As a consequence, we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems (such as levels in three dimensions and dynamic Euclidean minimum spanning trees in the plane). 1.1. Three surprises of high dimensions. This paper develops asymptotic methods to count faces of random high-dimensional polytopes; a seemingly dry and unpromising pursuit. Yet our conclusions have surprising implications - in statistics, probability, information theory, and signal processing - with potential impacts in practical subjects like medical imaging and digital communications. Before involving the reader in our lengthy analysis of high-dimensional face counting, we describe three implications of our results. 1.1.1. Convex Hulls of Gaussian Point Clouds. Consider a random point cloud of n points xi, i = 1, . . . , n, sampled independently and identically from a Gaussian distribution in R d with nonsingular covariance. This is a standard model of multivariate data; its properties are increasingly important in a wide range of applications. At the same time, it is an attractive and in some sense timeless object for theoretical study. Properties of the convex hull of the random point cloud X = xi have attracted interest for several decades, increasingly so in recent years; there is a nowvoluminous literature on the subject. The results could be significant for understanding outlier detection, or classification problems in machine learning. We study the convex hull of @math , the set of @math orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of @math is doubly spectrahedral, i.e., both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace. Our spectrahedral representations are explicit and are of minimum size, in the sense that there are no smaller spectrahedral representations of these convex bodies. We propose a general framework for nonparametric classification of multi-dimensional numerical patterns. Given training points for each class, it builds a set cover with convex sets each of which contains some training points of the class but no points of the other classes. Each convex set has thus an associated class label, and classification of a query point is made to the class of the convex set such that the projection of the query point onto its boundary is minimal. In this sense, the convex sets of a class are regarded as ''prototypes'' for that class. We then apply this framework to two special types of convex sets, minimum enclosing balls and convex hulls, giving algorithms for constructing a set cover with them and for computing the projection length onto their boundaries. For convex hulls, we also give a method for implicitly evaluating whether a point is contained in a convex hull, which can avoid computational difficulty for explicit construction of convex hulls in high-dimensional space. Motivated by the desire to cope with data imprecision Loffler (2009) [31], we study methods for taking advantage of preliminary information about point sets in order to speed up the computation of certain structures associated with them. In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon receiving a set P of n points, each of which lies on a distinct line of L, we can construct the convex hull of P efficiently. We show that in quadratic time and space it is possible to construct a data structure on L that enables us to compute the convex hull of any such point set P in O([email protected](n)log^@?n) expected time. If we further assume that the points are ''oblivious'' with respect to the data structure, the running time improves to O([email protected](n)). The same result holds when L is a set of line segments (in general position). We present several extensions, including a trade-off between space and query time and an output-sensitive algorithm. We also study the ''dual problem'' where we show how to efficiently compute the (= Non-negative matrix factorization (NMF) has recently received a lot of attention in data mining, information retrieval, and computer vision. It factorizes a non-negative input matrix V into two non-negative matrix factors V = WH such that W describes "clusters" of the datasets. Analyzing genotypes, social networks, or images, it can be beneficial to ensure V to contain meaningful cluster centroids'', i.e., to restrict W to be convex combinations of data points. But how can we run this convex NMF in the wild, i.e., given millions of data points? Triggered by the simple observation that each data point is a convex combination of vertices of the data convex hull, we propose to restrict W further to be vertices of the convex hull. The benefits of this convex-hull NMF approach are twofold. First, the expected size of the convex hull of the candidate set typically grows much slower than the data set. Second, distance preserving low-dimensional embeddings allow one to compute candidate vertices efficiently. Our extensive experimental evaluation shows that convex-hull NMF compares favorably to convex NMF for large data sets both in terms of speed and reconstruction quality. Moreover, we show that our method can easily be applied to large-scale, real-world data sets, in our case consisting of 1.6 million images respectively 160 million votes on World of Warcraft guilds. The anchor words algorithm performs provably efficient topic model inference by finding an approximate convex hull in a high-dimensional word co-occurrence space. However, the existing greedy algorithm often selects poor anchor words, reducing topic quality and interpretability. Rather than finding an approximate convex hull in a high-dimensional space, we propose to find an exact convex hull in a visualizable 2- or 3-dimensional space. Such low-dimensional embeddings both improve topics and clearly show users why the algorithm selects certain words. Finite convex geometries are combinatorial structures. It follows from a recent result of M. Richter and L.G. Rogers that there is an infinite set @math of planar convex polygons such that @math with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of @math to a finite subset in a natural way. An @math is a differentiable convex simple closed curve @math in the plane having an inscribed circle of radius @math and a circumscribed circle of radius @math such that the ratio @math is at least @math . Motivated by Richter and Rogers' result, we construct a set @math such that (1) @math contains all points of the plane as degenerate singleton circles and all of its non-singleton members are differentiable convex simple closed planar curves; (2) @math with respect to the geometric convex hull operator is a locally convex geometry; (3) as opposed to @math , @math is closed with respect to non-degenerate affine transformations; and (4) for every (small) positive @math and for every finite convex geometry, there are continuum many pairwise affine-disjoint finite subsets @math of @math such that each @math consists of almost-circles of accuracy @math and the convex geometry in question is represented by restricting the convex hull operator to @math . The affine-disjointness of @math and @math means that, in addition to @math , even @math is disjoint from @math for every non-degenerate affine transformation @math . We give novel algorithms for stochastic strongly-convex optimization in the gradient oracle model which return a O(1 T)-approximate solution after T iterations. The first algorithm is deterministic, and achieves this rate via gradient updates and historical averaging. The second algorithm is randomized, and is based on pure gradient steps with a random step size. his rate of convergence is optimal in the gradient oracle model. This improves upon the previously known best rate of O(log(T T), which was obtained by applying an online strongly-convex optimization algorithm with regret O(log(T)) to the batch setting. We complement this result by proving that any algorithm has expected regret of Ω(log(T)) in the online stochastic strongly-convex optimization setting. This shows that any online-to-batch conversion is inherently suboptimal for stochastic strongly-convex optimization. This is the first formal evidence that online convex optimization is strictly more difficult than batch stochastic convex optimization. Finding a point in the intersection of a collection of closed convex sets, that is the convex feasibility problem, represents the main modeling strategy for many computational problems. In this paper we analyze new stochastic reformulations of the convex feasibility problem in order to facilitate the development of new algorithmic schemes. We also analyze the conditioning problem parameters using certain (linear) regularity assumptions on the individual convex sets. Then, we introduce a general random projection algorithmic framework, which extends to the random settings many existing projection schemes, designed for the general convex feasibility problem. Our general random projection algorithm allows to project simultaneously on several sets, thus providing great flexibility in matching the implementation of the algorithm on the parallel architecture at hand. Based on the conditioning parameters, besides the asymptotic convergence results, we also derive explicit sublinear and linear convergence rates for this general algorithmic framework. We consider an algorithm that successively samples and minimizes stochastic models of the objective function. We show that under weak-convexity and Lipschitz conditions, the algorithm drives the expected norm of the gradient of the Moreau envelope to zero at the rate @math . Our result yields new complexity guarantees for the stochastic proximal point algorithm on weakly convex problems and for the stochastic prox-linear algorithm for minimizing compositions of convex functions with smooth maps. Moreover, our result also recovers the recently obtained complexity estimate for the stochastic proximal subgradient method on weakly convex problems. We discuss a new conceptual framework for the convexification of discrete optimization problems, and a general technique for obtaining approximations to the convex hull of the feasible set. The concepts come from disjunctive programming and the key tool is a description of the convex hull of a union of polyhedra in terms of a higher dimensional polyhedron. Although this description was known for several years, only recently was it shown by Jeroslow and Lowe to yield improved representations of discrete optimization problems. We express the feasible set of a discrete optimization problem as the intersection (conjunction) of unions of polyhedra, and define an operation that takes one such expression into another, equivalent one, with fewer conjuncts. We then introduce a class of relaxations based on replacing each conjunct (union of polyhedra) by its convex hull. The strength of the relaxations increases as the number of conjuncts decreases, and the class of relaxations forms a hierarchy that spans the spec...