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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 <jats:italic>X<\/jats:italic> is a set of polygonal curves in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {R}^d$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>R<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and the sets <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {R}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>R<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are metric balls defined by curve similarity metrics, such as the Fr\u00e9chet 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.<\/jats:p>","DOI":"10.1007\/s00454-021-00318-z","type":"journal-article","created":{"date-parts":[[2021,8,9]],"date-time":"2021-08-09T21:02:45Z","timestamp":1628542965000},"page":"1351-1381","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["The VC Dimension of Metric Balls under Fr\u00e9chet and Hausdorff Distances"],"prefix":"10.1007","volume":"66","author":[{"given":"Anne","family":"Driemel","sequence":"first","affiliation":[]},{"given":"Andr\u00e9","family":"Nusser","sequence":"additional","affiliation":[]},{"given":"Jeff M.","family":"Phillips","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5079-5003","authenticated-orcid":false,"given":"Ioannis","family":"Psarros","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,8,9]]},"reference":[{"key":"318_CR1","doi-asserted-by":"crossref","unstructured":"Afshani, P., Driemel, A.: On the complexity of range searching among curves (2017). arXiv:1707.04789","DOI":"10.1137\/1.9781611975031.58"},{"key":"318_CR2","doi-asserted-by":"crossref","unstructured":"Afshani, P., Driemel, A.: On the complexity of range searching among curves. 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