The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

Driemel, Anne; Nusser, André; Phillips, Jeff M.; Psarros, Ioannis

doi:10.1007/s00454-021-00318-z

Cited by 9 publications

(8 citation statements)

References 42 publications

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“…For several problems, we know how to bound the VC-dimension of the uniform function space, but not that of the non-uniform function space. Examples include the shortest-path metric in planar graphs [BT15] and the Fréchet distance [DNPP21]. Our new framework leads to new/improved coreset results for such clustering problems.…”

Section: Our Resultsmentioning

confidence: 99%

The Power of Uniform Sampling for Coresets

Braverman¹,

Cohen-Addad²,

-C.³

et al. 2022

Preprint

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Motivated by practical generalizations of the classic k-median and k-means objectives, such as clustering with size constraints, fair clustering, and Wasserstein barycenter, we introduce a meta-theorem for designing coresets for constrained-clustering problems. The meta-theorem reduces the task of coreset construction to one on a bounded number of ring instances with a much-relaxed additive error. This reduction enables us to construct coresets using uniform sampling, in contrast to the widely-used importance sampling, and consequently we can easily handle constrained objectives. Notably and perhaps surprisingly, this simpler sampling scheme can yield coresets whose size is independent of n, the number of input points.Our technique yields smaller coresets, and sometimes the first coresets, for a large number of constrained clustering problems, including capacitated clustering, fair clustering, Euclidean Wasserstein barycenter, clustering in minor-excluded graph, and polygon clustering under Fréchet and Hausdorff distance. Finally, our technique yields also smaller coresets for 1-median in lowdimensional Euclidean spaces, specifically of size Õ(ε −1.5 ) in R 2 and Õ(ε −1.6 ) in R 3 .

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Section: Our Resultsmentioning

confidence: 99%

The Power of Uniform Sampling for Coresets

Braverman¹,

Cohen-Addad²,

-C.³

et al. 2022

Preprint

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show abstract

“…So up to logarithmic factors these terms are bounded by each other. First Driemel et al [11] shows VC-dimension for a ground set of curves X m of length m, with respect to metric balls around curves of length k, for various distance between curves. The most relevant case is where m = 1 (so the ground set are points like Q), and the Hausdorff distance is considered, where the VC-dimension in d = 2 is bounded O(k 2 log(km)) = O(k 2 log k) and is at least Ω(max{k, log m}) = Ω(k).…”

Section: Strong Coresets For the Distance Between Trajectoriesmentioning

confidence: 99%

Sketched MinDist

Phillips,

Tang

2019

Preprint

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We sketch vectors of geometric objects J through the MinDist functionCollecting the vector of these sketch values induces a simple, effective, and powerful distance: the Euclidean distance between these sketched vectors. This paper shows how large this set Q needs to be under a variety of shapes and scenarios. For hyperplanes we provide direct connection to the sensitivity sampling framework, so relative error can be preserved in d dimensions using Q = O(d/ε 2 ). However, for other shapes, we show we need to enforce a minimum distance parameter ρ, and a domain size L. For d = 2 the sample size Q then can be Õ((L/ρ) • 1/ε 2 ). For objects (e.g., trajectories) with at most k pieces this can provide stronger for all approximations with Õ((L/ρ) • k 3 /ε 2 ) points. Moreover, with similar size bounds and restrictions, such trajectories can be reconstructed exactly using only these sketch vectors.

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“…Recent years have seen a raised interest in proximity data structures for trajectory analysis under the Fréchet distance [15,24,33,7,20,12,2,18,6,19,22,23,16,5]. An intuitive definition of the Fréchet distance uses the metaphor of a person walking a dog.…”

Section: Introductionmentioning

confidence: 99%

$(2+ε)$-ANN for time series under the Fréchet distance

Driemel,

Psarros

2020

Preprint

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We give the first ANN-data structure for time series under the continuous Fréchet distance, where ANN stands for approximate near neighbor. Given a parameter ∈ (0, 1], the data structure can be used to preprocess n curves in Euclidean R (aka time series), each of complexity m, to answer queries with a curve of complexity k by either returning a curve that lies within Fréchet distance 2 + , or answering that there exists no curve in the input within distance 1. In both cases, the answer is correct.Our data structure uses space in n • O −1 k + O(nm) and query time in O (k). The data structure is therefore especially useful in the asymmetric setting, where k m. This is wellmotivated for the continuous Fréchet distance as the distance measure takes into account the interior points along the edges of the curve.We show that the approximation factor achieved by our data structure is optimal in the cell-probe model of computation. Concretely, we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by O(L 1− ) for some constant > 0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L Ω(k) . Our data structure uses only a constant number of probes per query and does not have any dependency on L.We apply similar techniques for proving cell-probe lower bounds for the discrete Fréchet distance matching known upper bounds.

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The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

Cited by 9 publications

References 42 publications

The Power of Uniform Sampling for Coresets

The Power of Uniform Sampling for Coresets

Sketched MinDist

$(2+ε)$-ANN for time series under the Fréchet distance

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