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The Quickhull algorithm for convex hulls
 ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
, 1996
"... 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 twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond Algorithm. It is similar to the randomized, incremental algo ..."
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Cited by 452 (0 self)
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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 twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond 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 uses less memory. Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floatingpoint arithmetic, this assumption can lead to serious 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.
A Library for Doing Polyhedral Operations
, 1993
"... Polyhedra are geometric representations of linear systems of equations and inequalities. Since polyhedra are used to represent the iteration domains of nested loop programs, procedures for operating on polyhedra are useful for doing loop transformations and other program restructuring transformatio ..."
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Cited by 107 (13 self)
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Polyhedra are geometric representations of linear systems of equations and inequalities. Since polyhedra are used to represent the iteration domains of nested loop programs, procedures for operating on polyhedra are useful for doing loop transformations and other program restructuring transformations which are needed in parallelizing compilers. Thus a need for a library of polyhedral operations has recently been recognized in the parallelizing compiler community. Polyhedra are also used in the definition of domains of variables in systems of affine recurrence equations (SARE). Alpha is a language which is based on the SARE formalism in which all variables are declared over finite unions of polyhedra. This report describes a library of polyhedral functions which was developed to support the Alpha language environment, and which is general enough to satisfy the needs of researchers doing parallelizing compilers. This report describes the data structures used to represent domains, gives...
On The Expected Complexity Of The 3Dimensional Voronoi Diagram
 National Inst. of Standards and Technology
, 1990
"... Let S be a set of n sites chosen independently from a uniform distribution in a cube in 3 dimensional Euclidean space. In this paper, work by Bentley, Weide and Yao is extended to show that the Voronoi diagram for S has an expected O(n) number of faces. A consequence of the proof of this result is t ..."
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Cited by 5 (2 self)
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Let S be a set of n sites chosen independently from a uniform distribution in a cube in 3 dimensional Euclidean space. In this paper, work by Bentley, Weide and Yao is extended to show that the Voronoi diagram for S has an expected O(n) number of faces. A consequence of the proof of this result is that the Voronoi diagram for S can be constructed in expected O(n) time. 1. INTRODUCTION Consider a set S = fp 1 ; : : : ; p n g of n sites in d dimensional Euclidean space E d . The Voronoi diagram for S is a sequence V (p 1 ), : : : , V (p n ) of convex polyhedra covering E d , where for each i, i = 1; : : : ; n, V (p i ) is the Voronoi polyhedron of p i relative to S, i. e. the set of all points x in the space such that p i is as close to x as is any other site in S. The Voronoi diagram is an important geometrical concept that is used for solving a large number of problems in many areas. Accordingly, several algorithms have been devised and implemented for constructing it in two and...
Optimal Search for Minimum Error Rate Training
"... Minimum error rate training is a crucial component to many stateoftheart NLP applications, such as machine translation and speech recognition. However, common evaluation functions such as BLEU or word error rate are generally highly nonconvex and thus prone to search errors. In this paper, we pr ..."
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Cited by 1 (0 self)
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Minimum error rate training is a crucial component to many stateoftheart NLP applications, such as machine translation and speech recognition. However, common evaluation functions such as BLEU or word error rate are generally highly nonconvex and thus prone to search errors. In this paper, we present LPMERT, an exact search algorithm for minimum error rate training that reaches the global optimum using a series of reductions to linear programming. Given a set of Nbest lists produced from S input sentences, this algorithm finds a linear model that is globally optimal with respect to this set. We find that this algorithm is polynomial in N and in the size of the model, but exponential in S. We present extensions of this work that let us scale to reasonably large tuning sets (e.g., one thousand sentences), by either searching only promising regions of the parameter space, or by using a variant of LPMERT that relies on a beamsearch approximation. Experimental results show improvements over the standard Och algorithm. 1
Prioritized independent contact regions for form closure grasps
 in Proc. of the IEEE/RSJ Int. Conf. on Intelligent Robots and Systems
, 2011
"... Abstract — The concept of independent contact regions on a target object’s surface, in order to compensate for shortcomings in the positioning accuracy of robotic grasping devices, is well known. However, the numbers and distributions of contact points forming such regions is not unique and depends ..."
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Cited by 1 (1 self)
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Abstract — The concept of independent contact regions on a target object’s surface, in order to compensate for shortcomings in the positioning accuracy of robotic grasping devices, is well known. However, the numbers and distributions of contact points forming such regions is not unique and depends on the underlying computational method. In this work we present a computation scheme allowing to prioritize contact points for inclusion in the independent regions. This enables a user to affect their shape in order to meet the demands of the targeted application. The introduced method utilizes frictionless contact constraints and is able to efficiently approximate the space of disturbances resistible by all grasps comprising contacts within the independent regions. I.
Convex Hulls as an Hypothesis Language Bias
, 2004
"... Classification learning is dominated by systems which induce large numbers of small axisorthogonal decision surfaces which biases such systems towards particular hypothesis types. However, there is reason believe that many domains have underlying concepts which do not involve axis orthogonal surfac ..."
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Classification learning is dominated by systems which induce large numbers of small axisorthogonal decision surfaces which biases such systems towards particular hypothesis types. However, there is reason believe that many domains have underlying concepts which do not involve axis orthogonal surfaces. Further, the multiplicity of small decision regions mitigates against any holistic appreciation of the theories produced by these systems, notwithstanding the fact that many of the small regions are individually comprehensible. We propose the use of less strongly biased hypothesis languages which might be expected to model concepts using a number of structures close to the number of actual structures in the domain. An instantiation of such a language, a convex hull based classifier, CH1, has been implemented to investigate modeling concepts as a small number of large geometric structures in ndimensional space. A comparison of the number of regions induced is made against other wellknown systems on a representative selection of largely or wholly continuous valued machine learning tasks. The convex hull system is shown to produce a number of induced regions about an order of magnitude less than wellknown systems and very close to the number of actual concepts. This representation, as convex hulls, allows the possibility of extraction of higher level mathematical descriptions of the induced concepts, using the techniques of computational geometry.
Smooth Data Modelling and . . .
"... On the basis of studies of the olfactory bulb of a rabbit Freeman suggested that in the rest state the dynamics of this neural cluster is chaotic, but that when a familiar scent is presented the neural system rapidly simplifies its behaviour and the dynamics becomes more orderly, more nearly periodi ..."
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On the basis of studies of the olfactory bulb of a rabbit Freeman suggested that in the rest state the dynamics of this neural cluster is chaotic, but that when a familiar scent is presented the neural system rapidly simplifies its behaviour and the dynamics becomes more orderly, more nearly periodic than when in the rest state. This suggests an interesting model of recognition in biological neural systems. To realise this in an artificial neural system, some form of control of the chaotic neural behaviour is necessary to achieve periodic dynamical behaviour when a stimulus is presented. In this thesis we first study the general problem of modelling smooth systems and introduce a number of useful techniques relevant to the problem of modelling chaotic dynamics. After a preliminary review of chaotic dynamical systems and their control, and discussing several examples of neural chaos, we then construct a chaotic neural model. We show how this model can be successfully controlled using several different parametric control methods. However, such methods of control are external to the network and we are interested in the control of higher dimensional networks using a technique which is intrinsic to the neural dynamics. Using a higher dimensional system we investigate several methods of control and conclude that