Smallest enclosing ball for probabilistic data

Munteanu, Alexander; Sohler, Christian; Feldman, Dan

doi:10.1145/2582112.2582114

Cited by 22 publications

(8 citation statements)

References 32 publications

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“…There exists a PTAS for the stochastic minimum j-flat-center problem, under either the existential or the locational uncertainty model. This result also generalizes the PTAS for stochastic minimum enclosing ball (i.e., 0-flat-center) by Munteanu et al [32]. It also generalizes a previous PTAS for the stochastic minimum enclosing cylinder (i.e., 1-flat-center) problem in the existential model where the existential probability of each point is assumed to be lower bounded by a small fixed constant [25].Our techniques: Our techniques for both problems heavily rely on the powerful notion of coresets.…”

supporting

confidence: 85%

“…We remark that our overall approach is very different from that in Munteanu et al [32] (except one aforementioned step and that they also crucially used some machinary from the coreset literature). Munteanu et al [32] defined a near-metric distance measure m(A, B) = max a∈A,b∈B d(a, b) for two non-empty point sets A, B. This near-metric measure satisfies many metric properties, like non-negativity, symmetry and the triangle inequality.…”

Section: Stochastic Geometry Modelsmentioning

confidence: 96%

“…Roughly speaking, after linearization, we need to find a convex set K in a higher dimensional space such that the total probability of any point falling outside K is small, but not so small such that in each direction the expected directional width of P is comparable to that of K. Then, for those points inside K, it is possible to use a slight modification of the construction in [25] to construct a collection of weighted point sets. For the points outside K, since the total probability is small, we reduce the problem to a weighted j-flat-median problem, and use the coreset in [36] (this step is similar to that in [32]). By combining the two collections, we obtain the SJFC-CORESET S for the problem, which is of constant size.…”

Section: Stochastic Geometry Modelsmentioning

confidence: 99%

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Stochastic k-Center and j-Flat-Center Problems

Huang¹,

Li²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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Solving geometric optimization problems over uncertain data has become increasingly important in many applications and has attracted a lot of attentions in recent years. In this paper, we study two important geometric optimization problems, the k-center problem and the j-flat-center problem, over stochastic/uncertain data points in Euclidean spaces. For the stochastic k-center problem, we would like to find k points in a fixed dimensional Euclidean space, such that the expected value of the k-center objective is minimized. For the stochastic j-flat-center problem, we seek a j-flat (i.e., a j-dimensional affine subspace) such that the expected value of the maximum distance from any point to the j-flat is minimized. We consider both problems under two popular stochastic geometric models, the existential uncertainty model, where the existence of each point may be uncertain, and the locational uncertainty model, where the location of each point may be uncertain. We provide the first PTAS (Polynomial Time Approximation Scheme) for both problems under the two models. Our results generalize the previous results for stochastic minimum enclosing ball and stochastic enclosing cylinder.

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supporting

confidence: 85%

Section: Stochastic Geometry Modelsmentioning

confidence: 96%

Section: Stochastic Geometry Modelsmentioning

confidence: 99%

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Stochastic k-Center and j-Flat-Center Problems

Huang¹,

Li²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The first coresets for signal processing with applications to GPS or video data were suggested in [48,45,47]. The first results for probabilistic databases appeared recently [60,61] In this work we show that coreset paradigm can improve the computations described different sections, including provable guarantees of complexity in terms of training/inference time and memory, also for EEG applications. In particular, we demonstrate how coresets can be used to train a classifier in realtime for EEG signals.…”

Section: Related Work: Coresets For Eeg Real-time Processingmentioning

confidence: 63%

Real-time EEG classification via coresets for BCI applications

Netzer

Feldman

2020

Engineering Applications of Artificial Intelligence

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A brain-computer interface (BCI) based on the motor imagery (MI) paradigm translates one's motor intention into a control signal by classifying the Electroencephalogram (EEG) signal of different tasks. However, most existing systems either (i) use a high-quality algorithm to train the data off-line and run only classification in real-time, since the off-line algorithm is too slow, or (ii) use lowquality heuristics that are sufficiently fast for real-time training but introduces relatively large classification error.In this work, we propose a novel processing pipeline that allows real-time and parallel learning of EEG signals using high-quality but possibly inefficient algorithms. This is done by forging a link between BCI and core-sets, a technique that originated in computational geometry for handling streaming data via data summarization.We suggest an algorithm that maintains the representation such coreset tailored to handle the EEG signal which enables: (i) real time and continuous computation of the Common Spatial Pattern (CSP) feature extraction method on a coreset representation of the signal (instead on the signal itself) , (ii) improvement of the CSP algorithm efficiency with provable guarantees by applying CSP algorithm on the coreset, and (iii) real time addition of the data trials (EEG data windows) to the coreset.For simplicity, we focus on the CSP algorithm, which is a classic algorithm.Nevertheless, we expect that our coreset will be extended to other algorithms in future papers. In the experimental results we show that our system can indeed learn EEG signals in real-time for example a 64 channels setup with hundreds of time samples per second. Full open source is provided to reproduce the experiment and in the hope that it will be used and extended to more coresets and BCI applications in the future.

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“…The first inequality holds due to the concavity of µ (or equivalently, the convexity of µ −1 ): Hg(e) can be thought as the expected contribution of huge values of e. We need the following observation in [14]: the contribution of the huge values can be essentially linearized and separated from the contribution of normal values, in the sense of the following lemma. We note that the simple insight has been used in a variety of contexts in stochastic optimization problems (e.g., [47,33,34]).…”

Section: Class C Concavementioning

confidence: 99%

Maximizing Expected Utility for Stochastic Combinatorial Optimization Problems

Deshpande

2018

Mathematics of OR

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We study the stochastic versions of a broad class of combinatorial problems where the weights of the elements in the input dataset are uncertain. The class of problems that we study includes shortest paths, minimum weight spanning trees, and minimum weight matchings, and other combinatorial problems like knapsack. We observe that the expected value is inadequate in capturing different types of risk-averse or risk-prone behaviors, and instead we consider a more general objective which is to maximize the expected utility of the solution for some given utility function, rather than the expected weight (expected weight becomes a special case). Under the assumption that there is a pseudopolynomial time algorithm for the exact version of the problem (This is true for the problems mentioned above), 1 we can obtain the following approximation results for several important classes of utility functions:1. If the utility function µ is continuous, upper-bounded by a constant and lim x→+∞ µ(x) = 0, we show that we can obtain a polynomial time approximation algorithm with an additive error for any constant > 0.2. If the utility function µ is a concave increasing function, we can obtain a polynomial time approximation scheme (PTAS).3. If the utility function µ is increasing and has a bounded derivative, we can obtain a polynomial time approximation scheme.Our results recover or generalize several prior results on stochastic shortest path, stochastic spanning tree, and stochastic knapsack. Our algorithm for utility maximization makes use of the separability of exponential utility and a technique to decompose a general utility function into exponential utility functions, which may be useful in other stochastic optimization problems. 1 Following the literature [55], we differentiate between exact version and deterministic version of a problem; in the exact version of the problem, we are given a target value and asked to find a solution (e.g., a path) with exactly that value (i.e., path length).

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Smallest enclosing ball for probabilistic data

Cited by 22 publications

References 32 publications

Stochastic k-Center and j-Flat-Center Problems

Stochastic k-Center and j-Flat-Center Problems

Real-time EEG classification via coresets for BCI applications

Maximizing Expected Utility for Stochastic Combinatorial Optimization Problems

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