Learning random points from geometric graphs or orderings
Let Xv for v∈V be a family of n iid uniform points in the square 𝒮n=−n/2,n/22. Suppose first that we are given the random geometric graph G∈G(n,r), where vertices u and v are adjacent when the Euclidean distance dE(Xu,Xv) is at most r. Let n3/14≪r≪n1/2. Given G (without geometric information), in po...
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Published in | Random structures & algorithms Vol. 57; no. 2; pp. 339 - 370 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
New York
John Wiley & Sons, Inc
01.09.2020
Wiley Subscription Services, Inc Wiley |
Subjects | |
Online Access | Get full text |
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Summary: | Let Xv for v∈V be a family of n iid uniform points in the square 𝒮n=−n/2,n/22. Suppose first that we are given the random geometric graph G∈G(n,r), where vertices u and v are adjacent when the Euclidean distance dE(Xu,Xv) is at most r. Let n3/14≪r≪n1/2. Given G (without geometric information), in polynomial time we can with high probability approximately reconstruct the hidden embedding, in the sense that “up to symmetries,” for each vertex v we find a point within distance about r of Xv; that is, we find an embedding with “displacement” at most about r. Now suppose that, instead of G we are given, for each vertex v, the ordering of the other vertices by increasing Euclidean distance from v. Then, with high probability, in polynomial time we can find an embedding with displacement O(logn). |
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Bibliography: | Funding information This research was supported by the grants GRAMM, TIN2017‐86727‐C2‐1‐R (J.D.), and by IDEXLYON of Université de Lyon (Programme Investissements d'Avenir, ANR16‐IDEX‐0005), and by LABEX MILYON (ANR‐10‐LABX‐0070) of Université de Lyon, within the program “Investissements d'Avenir” (ANR‐11‐IDEX‐0007) (D.M.) |
ISSN: | 1042-9832 1098-2418 |
DOI: | 10.1002/rsa.20922 |