Epsilon-Coresets for Clustering (with Outliers) in Doubling Metrics

Huang, Like; Jiang, Shaofeng H.-C.; Li, Jian; Wu, Xuan

doi:10.1109/focs.2018.00082

Cited by 47 publications

(79 citation statements)

References 46 publications

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“…In general, a bounded doubling dimension does not imply a bounded VC dimension of the induced range space and vice versa. Recently, Huang et al [34] showed that if we allow a small (1 + ε)distortion of the distance function d, the shattering dimension can be upper bounded by O(ε −O(ddim(M,d)) ). It is conceivable that the doubling dimension of the metric space of the discrete Fréchet distance and Hausdorff distance is bounded, as long as the underlying metric has bounded doubling dimension.…”

Section: Range Spaces Induced By Distance Measuresmentioning

confidence: 99%

The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

Driemel

Nusser

Phillips

et al. 2021

Discrete Comput Geom

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The Vapnik–Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set X is a set of polygonal curves in $$\mathbb {R}^d$$ R d and the sets $$\mathcal {R}$$ R are metric balls defined by curve similarity metrics, such as the Fréchet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper and lower bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.

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Section: Range Spaces Induced By Distance Measuresmentioning

confidence: 99%

The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

Driemel

Nusser

Phillips

et al. 2021

Discrete Comput Geom

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“…ε-coresets are very popular in the field of clustering, cf. [19,18,10,21] and they are becoming a topic in other fields, too, cf. [22,29,31,15,26].…”

Section: Introductionmentioning

confidence: 99%

Coresets for $(k, \ell)$-Median Clustering under the Fréchet Distance

Buchin¹,

Rohde²

2021

Preprint

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We present an algorithm for computing ε-coresets for (k, ℓ)-median clustering of polygonal curves in R d under the Fréchet distance. This type of clustering, which has recently drawn increasing popularity due to a growing number of applications, is an adaption of Euclidean k-median clustering: we are given a set of polygonal curves in R d and want to compute k median curves such that the sum of distances from the given curves to their closest median curve is minimal. Additionally, we restrict the complexity, i.e., the number of vertices, of the median curves to be at most ℓ each, to suppress overfitting, a problem specific for sequential data. Our algorithm has running-time linear in the number of curves, sub-quartic in their maximum complexity and quadratic in ε −1 . With high probability it returns ε-coresets of size quadratic in ε −1 , logarithmic in the number of given curves and logarithmic in their maximum complexity. We achieve this result by applying the improved ε-coreset framework by Braverman et al. to a generalized k-median problem over an arbitrary metric space. Later we combine this result with the recent result by Driemel et al. on the VC dimension of metric balls under the Fréchet distance. Furthermore, our framework yields ε-coresets for any generalized k-median problem where the range space induced by the open metric balls of the underlying space has bounded VC dimension, which is of independent interest. Finally, we show that our ε-coresets can be used to improve the running-time of an existing approximation algorithm for (1, ℓ)-median clustering, taking a further step in the direction of practical (k, ℓ)-median clustering algorithms.

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“…This analysis was proven powerful in various metric spaces, such as doubling spaces by Huang, Jiang, Li and Wu [HJLW18], graphs of bounded treewidth by Baker, Braverman, Huang, Jiang, Krauthgamer, Wu [BBH + 20] or the shortest-path metric of a graph excluding a fixed minor by Braverman, Jiang, Krauthgamer and Wu [BJKW21]. However, range spaces of even heavily constrained metrics do not necessarily have small VC-dimension (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…However, range spaces of even heavily constrained metrics do not necessarily have small VC-dimension (e.g. bounded doubling dimension does not imply bounded VC-dimension or vice versa [HJLW18,LL06]), and applying previous techniques requires heavy additional machinery to adapt the VC-dimension approach to them. Moreover, the bounds provided are far from the bound obtained for Euclidean spaces: their dependency in k is at least Ω(k 2 ), leaving a significant gap to the best lower bounds of Ω(k).…”

Section: Introductionmentioning

confidence: 99%

A New Coreset Framework for Clustering

Cohen-Addad,

Saulpic,

Schwiegelshohn

2021

Preprint

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Given a metric space, the (k, z)-clustering problem consists of finding k centers such that the sum of the of distances raised to the power z of every point to its closest center is minimized. This encapsulates the famous k-median (z = 1) and k-means (z = 2) clustering problems. Designing small-space sketches of the data that approximately preserves the cost of the solutions, also known as coresets, has been an important research direction over the last 15 years.In this paper, we present a new, simple coreset framework that simultaneously improves upon the best known bounds for a large variety of settings, ranging from Euclidean space, doubling metric, minor-free metric, and the general metric cases: with Γ = min(ε −2 +ε −z , kε −2 )polylog(ε −1 ), this framework constructs coreset with sizein doubling metrics, improving upon the recent breakthrough of [Huang, Jiang, Li, Wu, FOCS' 18], who presented a coreset with size O(kfor graphs with treewidth t, improving on [Baker, Braverman, Huang, Jiang, Krauthgamer, Wu, ICML'20], who presented a coreset of size O(k 2 t/ε 2 ) for z = 1.for shortest paths metrics of graphs excluding a fixed minor. This improves on [Braverman, Jiang, Krauthgamer, Wu, SODA'21], who presented a coreset of size O(k 2 /ε 4 ).• Size O(Γ • k log n) in general discrete metric spaces, improving on the results of [Feldman, Lamberg, STOC'11], who presented a coreset of size O(kε −2z log n log k).A lower bound of Ω( k log n ε ) for k-Median in general metric spaces [Baker, Braverman, Huang, Jiang, Krauthgamer, Wu, ICML'20] implies that in general metrics as well as metrics with doubling dimension d, our bounds are optimal up to a poly log(1/ε)/ε factor. For graphs with treewidth t, the lower bound of Ω kt ε of [Baker, Braverman, Huang, Jiang, Krauthgamer, Wu, ICML'20] shows that our bounds are optimal up to the same factor.

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Epsilon-Coresets for Clustering (with Outliers) in Doubling Metrics

Cited by 47 publications

References 46 publications

The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

Coresets for $(k, \ell)$-Median Clustering under the Fréchet Distance

A New Coreset Framework for Clustering

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