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11
Local and global sparse Gaussian process approximations
 Proceedings of Artificial Intelligence and Statistics (AISTATS
, 2007
"... Gaussian process (GP) models are flexible probabilistic nonparametric models for regression, classification and other tasks. Unfortunately they suffer from computational intractability for large data sets. Over the past decade there have been many different approximations developed to reduce this co ..."
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Gaussian process (GP) models are flexible probabilistic nonparametric models for regression, classification and other tasks. Unfortunately they suffer from computational intractability for large data sets. Over the past decade there have been many different approximations developed to reduce this cost. Most of these can be termed global approximations, in that they try to summarize all the training data via a small set of support points. A different approach is that of local regression, where many local experts account for their own part of space. In this paper we start by investigating the regimes in which these different approaches work well or fail. We then proceed to develop a new sparse GP approximation which is a combination of both the global and local approaches. Theoretically we show that it is derived as a natural extension of the framework developed by Quiñonero Candela and Rasmussen [2005] for sparse GP approximations. We demonstrate the benefits of the combined approximation on some 1D examples for illustration, and on some large realworld data sets. 1
A framework for evaluating approximation methods for Gaussian process regression
 Journal of Machine Learning Research
"... Gaussian process (GP) predictors are an important component of many Bayesian approaches to machine learning. However, even a straightforward implementation of Gaussian process regression (GPR) requires O(n2) space and O(n3) time for a dataset of n examples. Several approximation methods have been pr ..."
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Gaussian process (GP) predictors are an important component of many Bayesian approaches to machine learning. However, even a straightforward implementation of Gaussian process regression (GPR) requires O(n2) space and O(n3) time for a dataset of n examples. Several approximation methods have been proposed, but there is a lack of understanding of the relative merits of the different approximations, and in what situations they are most useful. We recommend assessing the quality of the predictions obtained as a function of the compute time taken, and comparing to standard baselines (e.g., Subset of Data and FITC). We empirically investigate four different approximation algorithms on four different prediction problems, and make our code available to encourage future comparisons.
Placement and distributed deployment of sensor teams for triangulation based localization
 Proc. IEEE Int. Conf. on Robotics and Automation
"... ..."
Skinning arbitrary deformations
 In I3D ’07
, 2007
"... first determines the proxyjoints and their influences (c) and then computes the joint transformations, whose application in matrix palette skinning (d) gives a good approximation of the input deformation. Even though (b) and (d) appear to be almost identical, (d) needs about 17 times less memory th ..."
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Cited by 7 (1 self)
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first determines the proxyjoints and their influences (c) and then computes the joint transformations, whose application in matrix palette skinning (d) gives a good approximation of the input deformation. Even though (b) and (d) appear to be almost identical, (d) needs about 17 times less memory than (b) and can be rendered efficiently using the popular skinning algorithms. Matrix palette skinning (also known as skeletal subspace deformation) is a very popular realtime animation technique. So far, it has only been applied to the class of quasiarticulated objects, such as moving human or animal figures. In this paper, we demonstrate how to automatically construct skinning approximations of arbitrary precomputed animations, such as those of cloth or elastic materials. In contrast to previous approaches, our method is particularly well suited to input animations without rigid components. Our transformation fitting algorithm finds optimal skinning transformations (in a leastsquares sense) and therefore achieves considerably higher accuracy for nonquasiarticulated objects than previous methods. This allows the advantages of skinned animations (e.g., efficient rendering, restpose editing and fast collision detection) to be exploited for arbitrary deformations.
Complexity of Hypergraph Coloring and Seidel's Switching
, 2003
"... Seidel's switching of a vertex in a given graph results in making the vertex adjacent to precisely those vertices it was nonadjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching equivalent if one can be transformed into the other one by a sequence o ..."
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Cited by 4 (0 self)
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Seidel's switching of a vertex in a given graph results in making the vertex adjacent to precisely those vertices it was nonadjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching equivalent if one can be transformed into the other one by a sequence of Seidel's switchings. We consider the computational complexity of deciding if an input graph can be switched into a graph having a desired graph property. Among other results we show that switching to a regular graph is NPcomplete. The proof
Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph
, 2013
"... A standard way to approximate the distance between any two vertices p and q on a mesh is to compute, in the associated graph, a shortest path from p to q that goes through one of k sources, which are wellchosen vertices. Precomputing the distance between each of the k sources to all vertices of the ..."
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A standard way to approximate the distance between any two vertices p and q on a mesh is to compute, in the associated graph, a shortest path from p to q that goes through one of k sources, which are wellchosen vertices. Precomputing the distance between each of the k sources to all vertices of the graph yields an efficient computation of approximate distances between any two vertices. One standard method for choosing k sources, which has been used extensively and successfully for isometryinvariant surface processing, is the socalled Farthest Point Sampling (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources. In this paper, we analyze the stretch factor FFPS of approximate geodesics computed using FPS, which is the maximum, over all pairs of distinct vertices, of their approximated distance over their geodesic distance in the graph. We show that FFPS can be bounded in terms of the minimal value F ∗ of the stretch factor obtained using an optimal placement of k sources as FFPS 6 2r2eF ∗ + 2r2e + 8re + 1, where re is the ratio of the lengths of the longest and the shortest edges of the graph. This provides some evidence explaining why farthest point sampling has been used successfully for isometryinvariant shape processing. Furthermore, we show that it is NPcomplete to find k sources that minimize the stretch factor.
A Framework for Evaluating Approximation Methods for Gaussian Process Regression
"... Gaussian process (GP) predictors are an important component of many Bayesian approaches to machine learning. However, even a straightforward implementation of Gaussian process regression (GPR) requires O(n 2) space and O(n 3) time for a data set of n examples. Several approximation methods have been ..."
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Gaussian process (GP) predictors are an important component of many Bayesian approaches to machine learning. However, even a straightforward implementation of Gaussian process regression (GPR) requires O(n 2) space and O(n 3) time for a data set of n examples. Several approximation methods have been proposed, but there is a lack of understanding of the relative merits of the different approximations, and in what situations they are most useful. We recommend assessing the quality of the predictions obtained as a function of the compute time taken, and comparing to standard baselines (e.g., Subset of Data and FITC). We empirically investigate four different approximation algorithms on four different prediction problems, and make our code available to encourage future comparisons.
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"... Robots operating in a workspace can localize themselves by querying nodes of a sensornetwork deployed in the same workspace. This paper addresses the problem of computing the minimum number and placement of sensors so that the localization uncertainty at every point in the workspace is less than a ..."
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Robots operating in a workspace can localize themselves by querying nodes of a sensornetwork deployed in the same workspace. This paper addresses the problem of computing the minimum number and placement of sensors so that the localization uncertainty at every point in the workspace is less than a given threshold. We focus on triangulation based state estimation where measurements from two sensors must be combined for an estimate. This problem is NPhard in its most general from. For the general version, we present a solution framework based on integer linear programming and demonstrate its application in a firetower placement task. Next, we study the special case of bearingonly localization and present an approximation algorithm with a constant factor performance guarantee. Note to Practitioners Sensor networks can provide robust and scalable solutions to the localization problem which arises in numerous automation tasks. A common method for localization is triangulation in which measurements from two sensors are combined to obtain the location of a target. In this work, we study the problem of finding the minimum number, and placement of sensors in such a way that the uncertainty in localization is bounded at every point in the workspace when triangulation is used for estimating the location of a target. We present an efficient geometric algorithm for bearingonly localization which can be used for the deployment of cameranetworks. We also present a generic framework for arbitrary uncertainty metrics, and demonstrate its utility in an application where watchtowers are deployed to detect forest fires.
Sublinear Time Algorithms for Metric Space Problems
, 2000
"... Abstract In this paper we give approximation algorithms for the following problems on metric spaces: Furthest Pair, kmedian, Minimum Routing Cost Spanning Tree, Multiple Sequence Alignment, Maximum Traveling Salesman Problem, Maximum Spanning Tree and Average Distance. The key property of our algor ..."
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Abstract In this paper we give approximation algorithms for the following problems on metric spaces: Furthest Pair, kmedian, Minimum Routing Cost Spanning Tree, Multiple Sequence Alignment, Maximum Traveling Salesman Problem, Maximum Spanning Tree and Average Distance. The key property of our algorithms is that their running times is linear in the number of points. As the full specification of an npoint metric space is of size \Theta (n2), the complexity of our algorithms is sublinear with respect to the input size. All previous algorithms (exact or approximate) for these problems have running times \Omega (n2). We believe that our techniques can be applied to get similar bounds for other problems.
Constructive Approximation of Parametric 2Manifolds
"... For the important class of parametrically defined surfaces, we present a constructive, adaptive approximation algorithm that relies only on evaluation of the shape operator at chosen points. Earlier approximations with nets provided no explicit method for selection of the points from the manifold ..."
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For the important class of parametrically defined surfaces, we present a constructive, adaptive approximation algorithm that relies only on evaluation of the shape operator at chosen points. Earlier approximations with nets provided no explicit method for selection of the points from the manifold. Both approaches utilize previous constraints on the distribution of sample points in the manifold to ensure topological correctness of the approximant. The restriction to parametric surfaces was chosen because they are pervasive in practical applications in geometric modeling, computer graphics and scientific visualization. For these applications, the explication of specific input meshes is crucial, as can now be accomplished. 1.