A PTAS for k-means clustering based on weak coresets

Feldman, Dan; Monemizadeh, Morteza; Sohler, Christian

doi:10.1145/1247069.1247072

Cited by 152 publications

(163 citation statements)

References 16 publications

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“…Furthermore, a relaxed notion of coresets has been introduced by Feldman et al in [12]. The definition of weak coresets differs slightly from the original definition of a coreset.…”

Section: Related Workmentioning

confidence: 99%

“…Our approach is similar in spirit to the approach from [8,9] for the metric k-median problem and the metric k-means problem, as well as to the approach from [12] for the Euclidean k-means problem.…”

Section: Our Techniquesmentioning

confidence: 99%

“…We call this a Γ-weak coreset. Weak coresets have already been used in [12]. However, our notion differs slightly from the previous definition.…”

Section: γ-Weak Coresetsmentioning

confidence: 99%

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Coresets and Approximate Clustering for Bregman Divergences

Ackermann¹,

Blömer²

2009

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the generalized k-median problem with respect to a Bregman divergence D φ . Given a finite set P ⊆ R d of size n, our goal is to find a set C of size k such that the sum of errors cost(P, C) = p∈P min c∈C D φ (p, c) is minimized. The Bregman k-median problem plays an important role in many applications, e.g. information theory, statistics, text classification, and speech processing. We give the first coreset construction for this problem for a large subclass of Bregman divergences, including important dissimilarity measures such as the Kullback-Leibler divergence and the Itakura-Saito divergence. Using these coresets, we give a (1 + )-approximation algorithm for the Bregman k-median problem with running time O dkn + d 2 2 ( k ) Θ(1) log k+2 n . This result improves over the previousely fastest known (1+ )-approximation algorithm from [1]. Unlike the analysis of most coreset constructions our analysis does not rely on the construction of -nets. Instead, we prove our results by purely combinatorial means.

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“…Furthermore, a relaxed notion of coresets has been introduced by Feldman et al in [12]. The definition of weak coresets differs slightly from the original definition of a coreset.…”

Section: Related Workmentioning

confidence: 99%

Section: Our Techniquesmentioning

confidence: 99%

See 1 more Smart Citation

Coresets and Approximate Clustering for Bregman Divergences

Ackermann¹,

Blömer²

2009

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Usually v is proportional to the number of parameters that are needed to represent a dictionary D of D, which is, in the general case, the number dk of entries in the matrix D. In this section we consider the following problem. The input is a parameter k ≥ 1, a d × n matrix Y , and an error function err(·, [7][8][9], using coresets techniques.…”

Section: Coreset For All K-dictionary Queriesmentioning

confidence: 99%

“…When there are too many coresets in memory, a coreset for the union of coresets is constructed and the original coresets are deleted. See details in [8]. After constructing such a coreset C of size c = 10000 for all the columns of Y , we apply the K−SVD on the coreset using sparsity j = 10, and k = 256 atoms, and 40 iterations.…”

Section: Coresets For High-definition Imagesmentioning

confidence: 99%

From High Definition Image to Low Space Optimization

Feigin

Feldman

Sochen

2012

Lecture Notes in Computer Science

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Abstract. Signal and image processing have seen in the last few years an explosion of interest in a new form of signal/image characterization via the concept of sparsity with respect to a dictionary. An active field of research is dictionary learning: Given a large amount of example signals/images one would like to learn a dictionary with much fewer atoms than examples on one hand, and much more atoms than pixels on the other hand. The dictionary is constructed such that the examples are sparse on that dictionary i.e each image is a linear combination of small number of atoms. This paper suggests a new computational approach to the problem of dictionary learning. We show that smart non-uniform sampling, via the recently introduced method of coresets, achieves excellent results, with controlled deviation from the optimal dictionary. We represent dictionary learning for sparse representation of images as a geometric problem, and illustrate the coreset technique by using it together with the K−SVD method. Our simulations demonstrate gain factor of up to 60 in computational time with the same, and even better, performance. We also demonstrate our ability to perform computations on larger patches and high-definition images, where the traditional approach breaks down.

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2012

Advanced Quantum Communications

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A PTAS for k-means clustering based on weak coresets

Cited by 152 publications

References 16 publications

Coresets and Approximate Clustering for Bregman Divergences

Coresets and Approximate Clustering for Bregman Divergences

From High Definition Image to Low Space Optimization

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