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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2016.1
URN: urn:nbn:de:0030-drops-66241
URL: https://drops.dagstuhl.de/opus/volltexte/2016/6624/
Backurs, Arturs ; Sidiropoulos, Anastasios
Constant-Distortion Embeddings of Hausdorff Metrics into Constant-Dimensional l_p Spaces
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Abstract
We show that the Hausdorff metric over constant-size pointsets in constant-dimensional Euclidean space admits an embedding into constant-dimensional l_{infinity} space with constant distortion. More specifically for any s,d>=1, we obtain an embedding of the Hausdorff metric over pointsets of size s in d-dimensional Euclidean space, into l_{\infinity}^{s^{O(s+d)}} with distortion s^{O(s+d)}. We remark that any metric space M admits an isometric embedding into l_{infinity} with dimension proportional to the size of M. In contrast, we obtain an embedding of a space of infinite size into constant-dimensional l_{infinity}.We further improve the distortion and dimension trade-offs by considering probabilistic embeddings of the snowflake version of the Hausdorff metric. For the case of pointsets of size s in the real line of bounded resolution, we obtain a probabilistic embedding into l_1^{O(s*log(s()} with distortion O(s).
BibTeX - Entry
@InProceedings{backurs_et_al:LIPIcs:2016:6624,
author = {Arturs Backurs and Anastasios Sidiropoulos},
title = {{Constant-Distortion Embeddings of Hausdorff Metrics into Constant-Dimensional l_p Spaces}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
pages = {1:1--1:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-018-7},
ISSN = {1868-8969},
year = {2016},
volume = {60},
editor = {Klaus Jansen and Claire Mathieu and Jos{\'e} D. P. Rolim and Chris Umans},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6624},
URN = {urn:nbn:de:0030-drops-66241},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.1},
annote = {Keywords: metric embeddings, Hausdorff metric, distortion, dimension}
}
| Keywords: | metric embeddings, Hausdorff metric, distortion, dimension | |
| Seminar: | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016) | |
| Issue date: | 2016 | |
| Date of publication: | 06.09.2016 |
