Adaptive Sampling for k-Means Clustering

Aggarwal, Ankit; Deshpande, Amit; Kannan, Ravi

doi:10.1007/978-3-642-03685-9_2

Cited by 98 publications

(133 citation statements)

References 16 publications

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“…It gives a constant approximation ratio and is based on local search ideas popular in the related area of design and analysis of algorithms for facility location problems, e.g., [6]. Recently, [2] improved the analysis of [5] and gave an adaptive sampling based algorithm with constant factor approximation to the optimal cost. In an effort to make adaptive sampling techniques more scalable, [8] introduced k-means which reduces the number of passes needed over the data and enables improved parallelization.…”

Section: Introductionmentioning

confidence: 99%

An Algorithm for Online K-Means Clustering

Liberty¹,

Sriharsha²,

Sviridenko³

2015

2016 Proceedings of the Eighteenth Workshop on Algorithm Engineering and Experiments (ALENEX)

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This paper shows that one can be competitive with the k-means objective while operating online. In this model, the algorithm receives vectors v 1 , . . . , v n one by one in an arbitrary order. For each vector v t the algorithm outputs a cluster identifier before receiving v t+1 . Our online algorithm generates O(k log n log γn) clusters whose expected k-means cost is O(W * log n). Here, W * is the optimal k-means cost using k clusters and γ is the aspect ratio of the data. The dependence on γ is shown to be unavoidable and tight. We also show that, experimentally, it is not much worse than kmeans++ while operating in a strictly more constrained computational model.

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Section: Introductionmentioning

confidence: 99%

An Algorithm for Online K-Means Clustering

Liberty¹,

Sriharsha²,

Sviridenko³

2015

2016 Proceedings of the Eighteenth Workshop on Algorithm Engineering and Experiments (ALENEX)

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“…al. [4]. Also if we allow exponential running time dependence on k then we can also get a PTAS like in the work of Jaiswal et.…”

Section: K-means Clusteringmentioning

confidence: 99%

“…al. [4] we use weighted sampling to get a constant factor approximation to the k-means problem. Now we describe how all these steps can be implemented in sublinear time using the primitives that we described in the previous section.…”

Section: K-means Clusteringmentioning

confidence: 99%

“…To prove that this algorithm gets a constant factor approximation we use Theorem 1 and Theorem 4 of [4]. From the Theorem 1 of [4] and the fact that we sample from a distribution that is ε-close in total variation distance to the correct one we conclude that the set P that we chose satisfies Theorem 1 of [4] with probability of error at most ε 1 + O(k)δ.…”

mentioning

confidence: 99%

“…From the Theorem 1 of [4] and the fact that we sample from a distribution that is ε-close in total variation distance to the correct one we conclude that the set P that we chose satisfies Theorem 1 of [4] with probability of error at most ε 1 + O(k)δ. Then it is easy to see at Theorem 4 of [4] that when we have a constant factor approximation of the weights, we lose only a constant factor in the approximation ratio.…”

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confidence: 99%

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Faster Sublinear Algorithms using Conditional Sampling

Gouleakis¹,

Tzamos²,

Zampetakis³

2017

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

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A conditional sampling oracle for a probability distribution D returns samples from the conditional distribution of D restricted to a specified subset of the domain. A recent line of work [7,6] has shown that having access to such a conditional sampling oracle requires only polylogarithmic or even constant number of samples to solve distribution testing problems like identity and uniformity. This significantly improves over the standard sampling model where polynomially many samples are necessary.Inspired by these results, we introduce a computational model based on conditional sampling to develop sublinear algorithms with exponentially faster runtimes compared to standard sublinear algorithms. We focus on geometric optimization problems over points in high dimensional Euclidean space. Access to these points is provided via a conditional sampling oracle that takes as input a succinct representation of a subset of the domain and outputs a uniformly random point in that subset. We study two well studied problems: k-means clustering and estimating the weight of the minimum spanning tree. In contrast to prior algorithms for the classic model, our algorithms have time, space and sample complexity that is polynomial in the dimension and polylogarithmic in the number of points.Finally, we comment on the applicability of the model and compare with existing ones like streaming, parallel and distributed computational models. IntroductionConsider a scenario where you are given a dataset of input points X , from some domain Ω, stored in a random access memory and you want to estimate the number of distinct elements of this (multi-)set. One obvious way to do so is to iterate over all elements and use a hash table to find duplicates. Although simple, this solution becomes unattractive if the input is huge and it is too expensive to even parse it. In such cases, one natural goal is to get a good estimate of this number instead of computing it exactly. One way to do that is to pick some random points from X and estimate, based on those, the total number of distinct elements in the set. This is equivalent to getting samples from a probability distribution where the probability of each element is proportional to the number of times it appears in X . In the context of probability distributions, this is a well understood problem, called support estimation, and tight bounds are known for its sampling complexity. More specifically, in [22], it is shown that the number of samples needed is Θ(n/ log n) which, although sublinear, still has a huge dependence on the input size n = |X |.In several situations, more flexible access to the dataset might be possible, e.g. when data are stored in a database, which can significantly reduce the number of queries needed to perform support estimation or other tasks. One recent model, called conditional sampling, introduced by [7,6] for distribution testing, describes such a possibility. In that model, there is an underlying distribution D, and a conditional sampling oracle takes as input a ...

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Euclidean distance stratified random sampling based clustering model for big data mining

Pandey

Shukla

2021

Comp and Math Methods

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Big data mining is related to large‐scale data analysis and faces computational cost‐related challenges due to the exponential growth of digital technologies. Classical data mining algorithms suffer from computational deficiency, memory utilization, resource optimization, scale‐up, and speed‐up related challenges in big data mining. Sampling is one of the most effective data reduction techniques that reduces the computational cost, improves scalability and computational speed with high efficiency for any data mining algorithm in single and multiple machine execution environments. This study suggested a Euclidean distance‐based stratum method for stratum creation and a stratified random sampling‐based big data mining model using the K‐Means clustering (SSK‐Means) algorithm in a single machine execution environment. The performance of the SSK‐Means algorithm has achieved better cluster quality, speed‐up, scale‐up, and memory utilization against the random sampling‐based K‐Means and classical K‐Means algorithms using silhouette coefficient, Davies Bouldin index, Calinski Harabasz index, execution time, and speedup ratio internal measures.

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Adaptive Sampling for k-Means Clustering

Cited by 98 publications

References 16 publications

An Algorithm for Online K-Means Clustering

An Algorithm for Online K-Means Clustering

Faster Sublinear Algorithms using Conditional Sampling

Euclidean distance stratified random sampling based clustering model for big data mining

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