# Turning big data into tiny data: Constant-size coresets for k-means, PCA and projective clustering

@article{Feldman2013TurningBD, title={Turning big data into tiny data: Constant-size coresets for k-means, PCA and projective clustering}, author={Dan Feldman and Melanie Schmidt and Christian Sohler}, journal={ArXiv}, year={2013}, volume={abs/1807.04518} }

@d can be approximated up to (1 + e)-factor, for an arbitrary small e > 0, using the O(k/e2)-rank approximation of A and a constant. This implies, for example, that the optimal k-means clustering of the rows of A is (1 + e)-approximated by an optimal k-means clustering of their projection on the O(k/e2) first right singular vectors (principle components) of A.
A (j, k)-coreset for projective clustering is a small set of points that yields a (1 + e)-approximation to the sum of squared…

## 455 Citations

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The first such coreset of size independent of d is suggested, which is also the first deterministic coreset construction whose resulting size is not exponential in d.

### Tight Sensitivity Bounds For Smaller Coresets

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Experimental results on real-world datasets, including the English Wikipedia documents-term matrix, show that the bounds provided provide significantly smaller and data-dependent coresets also in practice.

### New Coresets for Projective Clustering and Applications

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This paper proposes the first algorithm that returns an L ∞ coreset of size polynomial in R d and gives the first strong coreset construction for general M -estimator regression, and provides experimental results based on real-world datasets, showing the eﬃcacy of the approach.

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- Mathematics, Computer Science2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
- 2018

The first strong coresets for the k-median and subspace approximation problems with sum of distances objective function are obtained, with a number of weighted points that is independent of both n and d; namely, their coresets have size poly(k/ε).

### Performance of Johnson-Lindenstrauss transform for k-means and k-medians clustering

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- 2019

This work shows that the cost of the optimal solution is preserved up to a factor of (1+ε) under a projection onto a random O(log(k /ε) / ε2)-dimensional subspace and that the bound on the dimension is nearly optimal.

### Near-optimal Coresets for Robust Clustering

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- 2022

This work constructs coresets for robust clustering in R d with m outliers by adapting to the outlier setting a recent framework, overcoming a new challenge that the participating terms in the cost, particularly the excluded m outlier points, are dependent on the center set C.

### Dimensionality Reduction for k-Means Clustering and Low Rank Approximation

- Computer ScienceSTOC
- 2015

This work shows 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+ε) error, and gives a simple alternative to known algorithms that has applications in the streaming setting.

### Dimensionality Reduction for the Sum-of-Distances Metric

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- 2021

A dimensionality reduction procedure to approximate the sum of distances of a given set of n points in R to any “shape” that lies in a k-dimensional subspace of R, and can be used to obtain poly(k/ε) size coresets for k-median and (k, 1)-subspace approximation problems in polynomial time.

### Towards optimal lower bounds for k-median and k-means coresets

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This paper achieves tight bounds for k-median in Euclidean spaces up to a factor O(ε−1 polylog k/ε), and is the first construction breaking through the ε−2· min(d,ε−2) barrier inherent in all previous coreset constructions.

### Coresets for clustering in Euclidean spaces: importance sampling is nearly optimal

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A unified two-stage importance sampling framework that constructs an ε-coreset for the (k,z)-clustering problem and relies on a new dimensionality reduction technique that connects two well-known shape fitting problems: subspace approximation and clustering, and may be of independent interest.

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