High-Dimensional Approximate r-Nets

The construction of r -nets offers a powerful tool in computational and metric geometry. We focus on high-dimensional spaces and present a new randomized algorithm which efficiently computes approximate r -nets with respect to Euclidean distance. For any fixed ϵ > 0 , the approximation factor is...

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Vydané v:	Algorithmica Ročník 82; číslo 6; s. 1675 - 1702
Hlavní autori:	Avarikioti, Z., Emiris, I. Z., Kavouras, L., Psarros, I.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.06.2020 Springer Nature B.V Springer Verlag
Predmet:	Algorithm Analysis and Problem Complexity Algorithms Clustering Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Euclidean geometry Geometry Mathematics Mathematics of Computing Metric Geometry Metric space Polynomials Software Theory of Computation General dimension Locality sensitive hashing Approximation algorithm nets Metric geometry
ISSN:	0178-4617, 1432-0541
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Abstract	The construction of r -nets offers a powerful tool in computational and metric geometry. We focus on high-dimensional spaces and present a new randomized algorithm which efficiently computes approximate r -nets with respect to Euclidean distance. For any fixed ϵ > 0 , the approximation factor is 1 + ϵ and the complexity is polynomial in the dimension and subquadratic in the number of points; the algorithm succeeds with high probability. Specifically, we improve upon the best previously known (LSH-based) construction of Eppstein et al. (Approximate greedy clustering and distance selection for graph metrics, 2015. CoRR arxiv: abs/1507.01555 ) in terms of complexity, by reducing the dependence on ϵ , provided that ϵ is sufficiently small. Moreover, our method does not require LSH but follows Valiant’s (J ACM 62(2):13, 2015. https://doi.org/10.1145/2728167 ) approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which r -nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the ( 1 + ϵ ) -approximate k -th nearest neighbor distance in time subquadratic in the size of the input.
AbstractList	The construction of r-nets offers a powerful tool in computational and metric geometry. We focus on high-dimensional spaces and present a new randomized algorithm which efficiently computes approximate r-nets with respect to Euclidean distance. For any fixed ϵ>0, the approximation factor is 1+ϵ and the complexity is polynomial in the dimension and subquadratic in the number of points; the algorithm succeeds with high probability. Specifically, we improve upon the best previously known (LSH-based) construction of Eppstein et al. (Approximate greedy clustering and distance selection for graph metrics, 2015. CoRR arxiv: abs/1507.01555) in terms of complexity, by reducing the dependence on ϵ, provided that ϵ is sufficiently small. Moreover, our method does not require LSH but follows Valiant’s (J ACM 62(2):13, 2015. 10.1145/2728167) approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which r-nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the (1+ϵ)-approximate k-th nearest neighbor distance in time subquadratic in the size of the input. The construction of r -nets offers a powerful tool in computational and metric geometry. We focus on high-dimensional spaces and present a new randomized algorithm which efficiently computes approximate r -nets with respect to Euclidean distance. For any fixed ϵ > 0 , the approximation factor is 1 + ϵ and the complexity is polynomial in the dimension and subquadratic in the number of points; the algorithm succeeds with high probability. Specifically, we improve upon the best previously known (LSH-based) construction of Eppstein et al. (Approximate greedy clustering and distance selection for graph metrics, 2015. CoRR arxiv: abs/1507.01555 ) in terms of complexity, by reducing the dependence on ϵ , provided that ϵ is sufficiently small. Moreover, our method does not require LSH but follows Valiant’s (J ACM 62(2):13, 2015. https://doi.org/10.1145/2728167 ) approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which r -nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the ( 1 + ϵ ) -approximate k -th nearest neighbor distance in time subquadratic in the size of the input. The construction of r-nets offers a powerful tool in computational and metric geometry. We focus on high-dimensional spaces and present a new randomized algorithm which efficiently computes approximate r-nets with respect to Euclidean distance. For any fixed ϵ>0, the approximation factor is 1+ϵ and the complexity is polynomial in the dimension and subquadratic in the number of points; the algorithm succeeds with high probability. Specifically, we improve upon the best previously known (LSH-based) construction of Eppstein et al. (Approximate greedy clustering and distance selection for graph metrics, 2015. CoRR arxiv: abs/1507.01555) in terms of complexity, by reducing the dependence on ϵ, provided that ϵ is sufficiently small. Moreover, our method does not require LSH but follows Valiant’s (J ACM 62(2):13, 2015. https://doi.org/10.1145/2728167) approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which r-nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the (1+ϵ)-approximate k-th nearest neighbor distance in time subquadratic in the size of the input.
Author	Avarikioti, Z. Psarros, I. Kavouras, L. Emiris, I. Z.
Author_xml	– sequence: 1 givenname: Z. surname: Avarikioti fullname: Avarikioti, Z. organization: School of Electrical and Computer Engineering, National Technical University of Athens – sequence: 2 givenname: I. Z. orcidid: 0000-0002-2339-5303 surname: Emiris fullname: Emiris, I. Z. organization: Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, ATHENA Research Center – sequence: 3 givenname: L. surname: Kavouras fullname: Kavouras, L. organization: School of Electrical and Computer Engineering, National Technical University of Athens – sequence: 4 givenname: I. orcidid: 0000-0002-5079-5003 surname: Psarros fullname: Psarros, I. email: ipsarros@di.uoa.gr organization: Department of Informatics and Telecommunications, National and Kapodistrian University of Athens
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Keywords	General dimension Locality sensitive hashing Approximation algorithm nets Metric geometry
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References	AndoniAIndykPNear-optimal hashing algorithms for approximate nearest neighbor in high dimensionsCommun. ACM200851111712210.1145/1327452.1327494 CoppersmithDRectangular matrix multiplication revisitedJ. Complex.19971314249144976010.1006/jcom.1997.04380872.68052 Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Geometric approximation via coresets. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry, MSRI, pp. 1–30. University Press (2005) Har-Peled, S., Mendel, M.: Fast construction of nets in low dimensional metrics, and their applications. In: Proceedings 21st Annual Symposium Computational Geometry, pp. 150–158 (2005). https://doi.org/10.1145/1064092.1064117 Anagnostopoulos, E., Emiris, I.Z., Psarros, I.: Low-quality dimension reduction and high-dimensional approximate nearest neighbor. CoRR arxiv: abs/1412.1683 (2014) ValiantGFinding correlations in subquadratic time, with applications to learning parities and the closest pair problemJ. ACM201562213334615210.1145/27281671333.68235 Alman, J., Chan, T.M., Williams, R.: Polynomial representations of threshold functions and algorithmic application. In: Proceedings 57th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 467–476 (2016) MitzenmacherMUpfalEProbability and Computing: Randomized Algorithms and Probabilistic Analysis2005CambridgeCambridge University Press10.1017/CBO97805118136031092.60001 Avarikioti, G., Emiris, I.Z., Kavouras, L., Psarros, I.: High-dimensional approximate r-nets. In: Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 16–30 (2017). https://doi.org/10.1137/1.9781611974782.2 Charikar, M.: Similarity estimation techniques from rounding algorithms. In: Proceedings 34th Annual ACM Symposium on Theory of Computing, 2002, Montréal, Canada, pp. 380–388 (2002) Eppstein, D., Har-Peled, S., Sidiropoulos, A.: Approximate greedy clustering and distance selection for graph metrics. CoRR arxiv: abs/1507.01555 (2015) Har-PeledSRaichelBNet and prune: a linear time algorithm for euclidean distance problemsJ. ACM201562644343722910.1145/28312301426.68273 Har-PeledSClustering motionDiscret. Comput. Geom.2004314545565205349810.1007/s00454-004-2822-71094.68103 Valiant, G.: Finding correlations in subquadratic time, with applications to learning parities and juntas. In: 53rd Annual IEEE Symposium Foundations of Computer Science (FOCS), pp. 11–20 (2012). https://doi.org/10.1109/FOCS.2012.27 Goel, A., Indyk, P., Varadarajan, K.: Reductions among high dimensional proximity problems. In: Proceedings 12th Symposium on Discrete Algorithms (SODA), pp. 769–778 (2001). http://dl.acm.org/citation.cfm?id=365411.365776. Accessed June 2016 DasguptaSGuptaAAn elementary proof of a theorem of Johnson and LindenstraussRandom Struct. Algorithms20032216065194385910.1002/rsa.100731018.51010 Avarikioti, G., Ryser, A., Wang, Y., Wattenhofer, R.: High dimensional clustering with r-nets. In: The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, pp. 3207–3214 (2019) S Har-Peled (664_CR12) 2004; 31 664_CR7 664_CR6 664_CR5 664_CR3 664_CR2 664_CR1 D Coppersmith (664_CR8) 1997; 13 S Dasgupta (664_CR9) 2003; 22 A Andoni (664_CR4) 2008; 51 S Har-Peled (664_CR14) 2015; 62 664_CR16 664_CR13 664_CR11 G Valiant (664_CR17) 2015; 62 M Mitzenmacher (664_CR15) 2005 664_CR10
References_xml	– reference: Avarikioti, G., Ryser, A., Wang, Y., Wattenhofer, R.: High dimensional clustering with r-nets. In: The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, pp. 3207–3214 (2019) – reference: Har-Peled, S., Mendel, M.: Fast construction of nets in low dimensional metrics, and their applications. In: Proceedings 21st Annual Symposium Computational Geometry, pp. 150–158 (2005). https://doi.org/10.1145/1064092.1064117 – reference: Avarikioti, G., Emiris, I.Z., Kavouras, L., Psarros, I.: High-dimensional approximate r-nets. In: Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 16–30 (2017). https://doi.org/10.1137/1.9781611974782.2 – reference: Har-PeledSClustering motionDiscret. Comput. Geom.2004314545565205349810.1007/s00454-004-2822-71094.68103 – reference: Charikar, M.: Similarity estimation techniques from rounding algorithms. In: Proceedings 34th Annual ACM Symposium on Theory of Computing, 2002, Montréal, Canada, pp. 380–388 (2002) – reference: Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Geometric approximation via coresets. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry, MSRI, pp. 1–30. University Press (2005) – reference: AndoniAIndykPNear-optimal hashing algorithms for approximate nearest neighbor in high dimensionsCommun. ACM200851111712210.1145/1327452.1327494 – reference: Eppstein, D., Har-Peled, S., Sidiropoulos, A.: Approximate greedy clustering and distance selection for graph metrics. CoRR arxiv: abs/1507.01555 (2015) – reference: Alman, J., Chan, T.M., Williams, R.: Polynomial representations of threshold functions and algorithmic application. In: Proceedings 57th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 467–476 (2016) – reference: CoppersmithDRectangular matrix multiplication revisitedJ. Complex.19971314249144976010.1006/jcom.1997.04380872.68052 – reference: Goel, A., Indyk, P., Varadarajan, K.: Reductions among high dimensional proximity problems. In: Proceedings 12th Symposium on Discrete Algorithms (SODA), pp. 769–778 (2001). http://dl.acm.org/citation.cfm?id=365411.365776. Accessed June 2016 – reference: ValiantGFinding correlations in subquadratic time, with applications to learning parities and the closest pair problemJ. ACM201562213334615210.1145/27281671333.68235 – reference: MitzenmacherMUpfalEProbability and Computing: Randomized Algorithms and Probabilistic Analysis2005CambridgeCambridge University Press10.1017/CBO97805118136031092.60001 – reference: Valiant, G.: Finding correlations in subquadratic time, with applications to learning parities and juntas. In: 53rd Annual IEEE Symposium Foundations of Computer Science (FOCS), pp. 11–20 (2012). https://doi.org/10.1109/FOCS.2012.27 – reference: DasguptaSGuptaAAn elementary proof of a theorem of Johnson and LindenstraussRandom Struct. Algorithms20032216065194385910.1002/rsa.100731018.51010 – reference: Anagnostopoulos, E., Emiris, I.Z., Psarros, I.: Low-quality dimension reduction and high-dimensional approximate nearest neighbor. CoRR arxiv: abs/1412.1683 (2014) – reference: Har-PeledSRaichelBNet and prune: a linear time algorithm for euclidean distance problemsJ. ACM201562644343722910.1145/28312301426.68273 – volume-title: Probability and Computing: Randomized Algorithms and Probabilistic Analysis year: 2005 ident: 664_CR15 doi: 10.1017/CBO9780511813603 – ident: 664_CR10 – ident: 664_CR11 – ident: 664_CR16 doi: 10.1109/FOCS.2012.27 – volume: 62 start-page: 13 issue: 2 year: 2015 ident: 664_CR17 publication-title: J. ACM doi: 10.1145/2728167 – ident: 664_CR7 doi: 10.1145/509907.509965 – volume: 62 start-page: 44 issue: 6 year: 2015 ident: 664_CR14 publication-title: J. ACM doi: 10.1145/2831230 – ident: 664_CR1 – volume: 13 start-page: 42 issue: 1 year: 1997 ident: 664_CR8 publication-title: J. Complex. doi: 10.1006/jcom.1997.0438 – volume: 51 start-page: 117 issue: 1 year: 2008 ident: 664_CR4 publication-title: Commun. ACM doi: 10.1145/1327452.1327494 – ident: 664_CR3 – ident: 664_CR6 doi: 10.1609/aaai.v33i01.33013207 – volume: 22 start-page: 60 issue: 1 year: 2003 ident: 664_CR9 publication-title: Random Struct. Algorithms doi: 10.1002/rsa.10073 – volume: 31 start-page: 545 issue: 4 year: 2004 ident: 664_CR12 publication-title: Discret. Comput. Geom. doi: 10.1007/s00454-004-2822-7 – ident: 664_CR2 doi: 10.1109/FOCS.2016.57 – ident: 664_CR5 doi: 10.1137/1.9781611974782.2 – ident: 664_CR13 doi: 10.1145/1064092.1064117
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Snippet	The construction of r -nets offers a powerful tool in computational and metric geometry. We focus on high-dimensional spaces and present a new randomized... The construction of r-nets offers a powerful tool in computational and metric geometry. We focus on high-dimensional spaces and present a new randomized...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Clustering Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Euclidean geometry Geometry Mathematics Mathematics of Computing Metric Geometry Metric space Polynomials Software Theory of Computation
Title	High-Dimensional Approximate r-Nets
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