Turning Big Data Into Tiny Data: Constant-Size Coresets for  $k$-Means, PCA, and Projective Clustering

Feldman, Dan; Schmidt, Melanie; Sohler, Christian

doi:10.1137/18m1209854

Cited by 77 publications

(45 citation statements)

References 27 publications

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“…This allows us to handle regularization terms or costs that are common in machine learning such as to compute X that minimizes cost f (P, X) + |X| over every set X of |X| ≤ k centers (P. K. Agarwal & Mustafa, 2004;Bachem, Lucic, & Krause, 2015). Same property occurs for other model parameters such as j-subspaces; (Feldman, Schmidt, & Sohler, 2013a).…”

Section: Why Coresets?mentioning

confidence: 67%

“…For example, the k-means problem is NP-hard when k is part of the input (Mahajan, Nimbhorkar, & Varadarajan, 2009). However, composable coresets of near-linear size in (k/ε) can be used to produce 1 ± ε multiplicative factor approximation in O(ndk) time, even for streaming distributed data in parallel (Braverman et al, 2016;Feldman, Schmidt, & Sohler, 2013a).…”

Section: Why Coresets?mentioning

confidence: 99%

“…Probabilistic constructions failed with a given probability δ ∈ (0, 1) and the coreset size usually depends poly-logarithmically on log(1/δ) (by Hoeffding's inequality (Hoeffding, 1963)), and rarely linearly on 1/δ (by Markov's inequality (Inaba, Katoh, & Imai, 1994)). First coresets were deterministic of size exponential in their (VC-)dimension d (P. Agarwal et al, 2005;Feldman, Fiat, & Sharir, 2006), random constructions whose output is polynomial in d replaced them (Chen, 2009a), and independency of d is sometimes possible using either weak coresets (Feldman & Tassa, 2015) or squared Euclidean distances (Feldman, Schmidt, & Sohler, 2013a;Barger & Feldman, 2016;Feldman, Ozer, & Rus, 2017).…”

Section: Coreset Typesmentioning

confidence: 99%

“…Low-dimensional coresets are coresets that instead of having small number of points, are contained in a low dimensional space in some sense. The classic examples are low-rank approximations (SVD/PCA) (Mahoney, 2011;Feldman, Schmidt, & Sohler, 2013a;Cohen, Elder, Musco, Musco, & Persu, 2015), JL-lemma (random projections) (Boutsidis, Zouzias, Mahoney, & Drineas, 2015) and well conditioned matrices (A. Dasgupta, Drineas, Harb, Kumar, & Mahoney, 2008).…”

Section: (Right)mentioning

confidence: 99%

“…Example: The VC-dimension for the range space of k balls in R d is O(dk log k); (Feldman, Schmidt, & Sohler, 2013a). However, a (1 + ε)-approximation to the optimal query (center) for the k-means/median/center problem is spanned by a convex combination of k/ε O(1) input points in P (Shyamalkumar & Varadarajan, 2007;Feldman, Ozer, & Rus, 2017).…”

Section: Query Setsmentioning

confidence: 99%

See 4 more Smart Citations

Core‐sets: An updated survey

Feldman

2019

WIREs Data Min & Knowl

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In optimization or machine learning problems we are given a set of items, usually points in some metric space, and the goal is to minimize or maximize an objective function over some space of candidate solutions. For example, in clustering problems, the input is a set of points in some metric space, and a common goal is to compute a set of centers in some other space (points, lines) that will minimize the sum of distances to these points. In database queries, we may need to compute such a some for a specific query set of k centers. However, traditional algorithms cannot handle modern systems that require parallel real‐time computations of infinite distributed streams from sensors such as GPS, audio or video that arrive to a cloud, or networks of weaker devices such as smartphones or robots. Core‐set is a “small data” summarization of the input “big data,” where every possible query has approximately the same answer on both data sets. Generic techniques enable efficient coreset maintenance of streaming, distributed and dynamic data. Traditional algorithms can then be applied on these coresets to maintain the approximated optimal solutions. The challenge is to design coresets with provable tradeoff between their size and approximation error. This survey summarizes such constructions in a retrospective way, that aims to unified and simplify the state‐of‐the‐art. This article is categorized under Algorithmic Development > Structure Discovery Fundamental Concepts of Data and Knowledge > Big Data Mining Technologies > Machine Learning Algorithmic Development > Scalable Statistical Methods

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Section: Why Coresets?mentioning

confidence: 67%

Section: Why Coresets?mentioning

confidence: 99%

Section: Coreset Typesmentioning

confidence: 99%

Section: (Right)mentioning

confidence: 99%

Section: Query Setsmentioning

confidence: 99%

See 3 more Smart Citations

Core‐sets: An updated survey

Feldman

2019

WIREs Data Min & Knowl

Self Cite

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Towards faster big data analytics for anti‐jamming applications in vehicular ad‐hoc network

Bangui

Bühnová

et al. 2021

Trans Emerging Tel Tech

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Nowadays, Wireless Vehicular Ad-Hoc Network (VANET) has become a valuable asset for transportation systems. However, this advanced technology is characterized by highly distributed and networked environment, which makes VANET communications vulnerable to malicious jamming attacks. Although Big Data Analytics has been used to solve this critical security issue by supporting the development of anti-jamming applications, as the amount of vehicular data is growing exponentially, the anti-jamming applications face many challenges (i., reactions in real-time) due to the lack of specific solutions that can keep up with the fast advancement of VANET. In this paper, we propose a new vehicular data prioritization model based on coresets to accelerate the Big Data Analytics in VANET. Our experimental evaluation shows that our solution can significantly increase the efficiency for clustering in jamming detection while keeping and improving the clustering quality. Also, the proposed solution can enable the real-time detection and be integrated to anti-jamming applications.Hind Bangui, Mouzhi Ge, Barbora Buhnova, and TRANG Le Hong contributed equally to this study.

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Summation pollution of principal component analysis and an improved algorithm for location sensitive data

Cai

2021

Numerical Linear Algebra App

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Principal component analysis (PCA) is widely used for dimensionality reduction and unsupervised learning. The reconstruction error is sometimes large even when a large number of eigenmode is used. In this paper, we show that this unexpected error source is the pollution effect of a summation operation in the objective function of the PCA algorithm. The summation operator brings together unrelated parts of the data into the same optimization and the result is the reduction of the accuracy of the overall algorithm. We introduce a domain decomposed PCA that improves the accuracy, and surprisingly also increases the parallelism of the algorithm. To demonstrate the accuracy and parallel efficiency of the proposed algorithm, we consider three applications including a face recognition problem, a brain tumor detection problem using two-and three-dimensional MRI images.

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Turning Big Data Into Tiny Data: Constant-Size Coresets for $k$-Means, PCA, and Projective Clustering

Cited by 77 publications

References 27 publications

Core‐sets: An updated survey

Core‐sets: An updated survey

Towards faster big data analytics for anti‐jamming applications in vehicular ad‐hoc network

Summation pollution of principal component analysis and an improved algorithm for location sensitive data

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