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Turning Big data into tiny data: Constant-size coresets for k-means, PCA and projective clustering
TL;DR: In this paper, the authors developed and analyzed a method to reduce the size of a very large set of data points in a high dimensional Euclidean space R d to a small set of weighted points such that the result of a predetermined data analysis task on the reduced set is approximately the same as that for the original point set.
Abstract: We develop and analyze a method to reduce the size of a very large set of data points in a high dimensional Euclidean space R d to a small set of weighted points such that the result of a predetermined data analysis task on the reduced set is approximately the same as that for the original point set. For example, computing the first k principal components of the reduced set will return approximately the first k principal components of the original set or computing the centers of a k-means clustering on the reduced set will return an approximation for the original set. Such a reduced set is also known as a coreset. The main new feature of our construction is that the cardinality of the reduced set is independent of the dimension d of the input space and that the sets are mergable. The latter property means that the union of two reduced sets is a reduced set for the union of the two original sets (this property has recently also been called composability, see Indyk et. al., PODS 2014). It allows us to turn our methods into streaming or distributed algorithms using standard approaches. For problems such as k-means and subspace approximation the coreset sizes are also independent of the number of input points. Our method is based on projecting the points on a low dimensional subspace and reducing the cardinality of the points inside this subspace using known methods. The proposed approach works for a wide range of data analysis techniques including k-means clustering, principal component analysis and subspace clustering. The main conceptual contribution is a new coreset definition that allows to charge costs that appear for every solution to an additive constant.
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TL;DR: This issue introduces four articles, written by influential scientists in the field, covering the most interesting and state-of-the-art topics on Big Data mining, and presents a broad overview of the topic, its current status, controversy, and a forecast to the future.
Abstract: Big Data is a new term used to identify datasets that we can not manage with current methodologies or data mining software tools due to their large size and complexity. Big Data mining is the capability of extracting useful information from these large datasets or streams of data. New mining techniques are necessary due to the volume, variability, and velocity, of such data. The Big Data challenge is becoming one of the most exciting opportunities for the years to come. We present in this issue, a broad overview of the topic, its current status, controversy, and a forecast to the future. We introduce four articles, written by influential scientists in the field, covering the most interesting and state-of-the-art topics on Big Data mining.
731 citations
TL;DR: An extensive literature review over the period 2002-2013 of machine learning methods that were used to address common issues in WSNs is presented and a comparative guide is provided to aid WSN designers in developing suitable machine learning solutions for their specific application challenges.
Abstract: Wireless sensor networks (WSNs) monitor dynamic environments that change rapidly over time. This dynamic behavior is either caused by external factors or initiated by the system designers themselves. To adapt to such conditions, sensor networks often adopt machine learning techniques to eliminate the need for unnecessary redesign. Machine learning also inspires many practical solutions that maximize resource utilization and prolong the lifespan of the network. In this paper, we present an extensive literature review over the period 2002–2013 of machine learning methods that were used to address common issues in WSNs. The advantages and disadvantages of each proposed algorithm are evaluated against the corresponding problem. We also provide a comparative guide to aid WSN designers in developing suitable machine learning solutions for their specific application challenges.
704 citations
TL;DR: The question that arises now is, how to develop a high performance platform to efficiently analyze big data and how to design an appropriate mining algorithm to find the useful things from big data.
Abstract: The age of big data is now coming. But the traditional data
analytics may not be able to handle such large quantities of data. The question that arises now is, how to develop a high performance platform to efficiently analyze big data and
how to design an appropriate mining algorithm to find the useful things from big data. To deeply discuss this issue, this paper begins with a brief introduction to data analytics, followed by the discussions of big data analytics.
Some important open issues and further research directions will also be presented for the next step of big data analytics.
604 citations
TL;DR: This paper investigates representative Big Data applications from typical services like finance & economics, healthcare, Supply Chain Management (SCM), and manufacturing sector and discusses current movements on the Big Data for SCM in service and manufacturing world-wide including North America, Europe, and Asia Pacific region.
412 citations
14 Jun 2015
TL;DR: In this paper, the authors show how to approximate a data matrix A with a much smaller sketch ~A that can be used to solve a general class of constrained k-rank approximation problems to within (1+e) error.
Abstract: We show how to approximate a data matrix A with a much smaller sketch ~A that can be used to solve a general class of constrained k-rank approximation problems to within (1+e) error. Importantly, this class includes k-means clustering and unconstrained low rank approximation (i.e. principal component analysis). By reducing data points to just O(k) dimensions, we generically accelerate any exact, approximate, or heuristic algorithm for these ubiquitous problems. For k-means dimensionality reduction, we provide (1+e) relative error results for many common sketching techniques, including random row projection, column selection, and approximate SVD. For approximate principal component analysis, we give a simple alternative to known algorithms that has applications in the streaming setting. Additionally, we extend recent work on column-based matrix reconstruction, giving column subsets that not only 'cover' a good subspace for A}, but can be used directly to compute this subspace.Finally, for k-means clustering, we show how to achieve a (9+e) approximation by Johnson-Lindenstrauss projecting data to just O(log k/e2) dimensions. This is the first result that leverages the specific structure of k-means to achieve dimension independent of input size and sublinear in k.
314 citations
References
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Proceedings Article•
03 Jan 2001TL;DR: This paper proposed a generative model for text and other collections of discrete data that generalizes or improves on several previous models including naive Bayes/unigram, mixture of unigrams, and Hof-mann's aspect model, also known as probabilistic latent semantic indexing (pLSI).
Abstract: We propose a generative model for text and other collections of discrete data that generalizes or improves on several previous models including naive Bayes/unigram, mixture of unigrams [6], and Hof-mann's aspect model, also known as probabilistic latent semantic indexing (pLSI) [3]. In the context of text modeling, our model posits that each document is generated as a mixture of topics, where the continuous-valued mixture proportions are distributed as a latent Dirichlet random variable. Inference and learning are carried out efficiently via variational algorithms. We present empirical results on applications of this model to problems in text modeling, collaborative filtering, and text classification.
25,546 citations
07 Jan 2007
TL;DR: By augmenting k-means with a very simple, randomized seeding technique, this work obtains an algorithm that is Θ(logk)-competitive with the optimal clustering.
Abstract: The k-means method is a widely used clustering technique that seeks to minimize the average squared distance between points in the same cluster. Although it offers no accuracy guarantees, its simplicity and speed are very appealing in practice. By augmenting k-means with a very simple, randomized seeding technique, we obtain an algorithm that is Θ(logk)-competitive with the optimal clustering. Preliminary experiments show that our augmentation improves both the speed and the accuracy of k-means, often quite dramatically.
7,539 citations
Proceedings Article•
01 Jan 2000TL;DR: Two different multiplicative algorithms for non-negative matrix factorization are analyzed and one algorithm can be shown to minimize the conventional least squares error while the other minimizes the generalized Kullback-Leibler divergence.
Abstract: Non-negative matrix factorization (NMF) has previously been shown to be a useful decomposition for multivariate data. Two different multiplicative algorithms for NMF are analyzed. They differ only slightly in the multiplicative factor used in the update rules. One algorithm can be shown to minimize the conventional least squares error while the other minimizes the generalized Kullback-Leibler divergence. The monotonic convergence of both algorithms can be proven using an auxiliary function analogous to that used for proving convergence of the Expectation-Maximization algorithm. The algorithms can also be interpreted as diagonally rescaled gradient descent, where the rescaling factor is optimally chosen to ensure convergence.
7,345 citations
Book•
01 Jan 2004TL;DR: This book provides an easy introduction for students and researchers to the growing field of kernel-based pattern analysis, demonstrating with examples how to handcraft an algorithm or a kernel for a new specific application, and covering all the necessary conceptual and mathematical tools to do so.
Abstract: Kernel methods provide a powerful and unified framework for pattern discovery, motivating algorithms that can act on general types of data (e.g. strings, vectors or text) and look for general types of relations (e.g. rankings, classifications, regressions, clusters). The application areas range from neural networks and pattern recognition to machine learning and data mining. This book, developed from lectures and tutorials, fulfils two major roles: firstly it provides practitioners with a large toolkit of algorithms, kernels and solutions ready to use for standard pattern discovery problems in fields such as bioinformatics, text analysis, image analysis. Secondly it provides an easy introduction for students and researchers to the growing field of kernel-based pattern analysis, demonstrating with examples how to handcraft an algorithm or a kernel for a new specific application, and covering all the necessary conceptual and mathematical tools to do so.
6,050 citations
TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.
Abstract: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady. The paper was first published in Russian as Вапник В. Н. and Червоненкис А. Я. О равномерноЙ сходимости частот появления событиЙ к их вероятностям. Теория вероятностеЙ и ее применения 16(2), 264–279 (1971).
3,939 citations