Fast Smallest-Enclosing-Ball Computation in High Dimensions

Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high-dimensional spaces. We consider the case where data are drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to "learn" the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide learning-theoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to self-intersection of the submanifold. We are also able to treat the situation where the data are "noisy" and lie near rather than on the submanifold in question.

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“…Fast algorithms for the smallest ball problem exist. See [12] for theoretical discussion and [14] for downloadable algorithms from the web. 4.…”

Section: Computing the Homology Of Umentioning

confidence: 99%

Finding the Homology of Submanifolds with High Confidence from Random Samples

Niyogi

Smale

Weinberger

2008

Discrete Comput Geom

342

106

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“…Exact algorithms of MEB Traditional exact MEB algorithms can not effectively deal with high-dimensional data. For example, prune-and-search algorithm proposed by Megiddo, heuristic-based move-to-front algorithm proposed by Welzl, quadratic programming method proposed by GÄartner, the F-G-K algorithm proposed by Fischer [16]. We briefly present the iteration strategy of F-G-K algorithm below, called the dropping and walking (see Figure 1).…”

Section: Review On Meb Algorithms a Formulation Of Mebmentioning

confidence: 99%

New Intelligent Classification Method Based On Improved Meb Algorithm

Wang

Liu

2014

International Journal on Smart Sensing and Intelligent Systems

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Abstract-We present a simple approximate algorithm to compute the Minimum Enclosing Ball (MEB) of training samples in high dimensional Euclidean space. We prove theoretically that the proposed algorithm converges to the optimum within any precision quickly. Compared to popular MEB algorithms, it has the competitive performances on both training time and accuracy. Besides, the proposed algorithm does not need any extra requirement on kernels, it can be linked with extensive kernel methods, consequently. We also use the proposed algorithm to handle Binary Classification, Multi-class Classification, and Image Clustering problems. Experiments on both synthetic and realworld data sets demonstrate the validity of the algorithm we proposed.

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“…Finding MEB is a well-studied computational geometry problem and efficient solutions exist for problems with up to a few hundred dimensions. We use an efficient implementation of Fischer et al [6] to compute MEBs. Finally, we normalize this intersection and convert it to a functional similarity score.…”

Section: Glu R1 R2mentioning

confidence: 99%

Functional similarities of reaction sets in metabolic pathways

Kahveci

2010

Proceedings of the First ACM International Conference on Bioinformatics and Computational Biology

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Analyzing metabolic pathways by means of their steady states has proven to be accurate and efficient for practical purposes. The models such as elementary flux modes (EFMs) and extreme pathways (EPs) define the boundaries of the metabolic flux cone that is the set of all steady states of a pathway. However, the contributions of the subsets of pathway components in this flux cone so far has not been characterized mathematically. Also, the functional similarities of different component sets (e.g., sets of reactions) has not been expressed as a function of the steady states of metabolic pathways. Here, we aim to fill this gap by proposing a model that quantifies the impact of a set of components on the steady states of a pathway by using EFMs. At a high level, we model the impact of a given component set as the change in the flux cone when all the elements of that set are inhibited. Furthermore, given two sets of components from different pathways, we measure their functional similarity as the similarity of their impacts on corresponding pathways. Computing this functional similarity is a computationally challenging task as it requires finding the volumes of the intersection and the union of two polyhedral cones in high dimensional space. These volumes cannot be expressed in closed form. In this paper, we develop a novel method that first transforms the polyhedral cones to polytopes and then uses minimum enclosing balls to approximate this intersection efficiently. Our experiments on real metabolic pathways demonstrate that our method is of great use for both measuring the impacts of pathway components and identifying functionally similar component sets.

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Fast Smallest-Enclosing-Ball Computation in High Dimensions

Cited by 90 publications

References 9 publications

Finding the Homology of Submanifolds with High Confidence from Random Samples

Finding the Homology of Submanifolds with High Confidence from Random Samples

New Intelligent Classification Method Based On Improved Meb Algorithm

Functional similarities of reaction sets in metabolic pathways

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