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  Phase transition in the family of p-resistances

Alamgir, M., & von Luxburg, U. (2012). Phase transition in the family of p-resistances. In J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, & K. Weinberger (Eds.), Advances in Neural Information Processing Systems 24 (pp. 379-387). Red Hook, NY, USA: Curran.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0013-B888-5 Version Permalink: http://hdl.handle.net/21.11116/0000-0001-194C-F
Genre: Conference Paper

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 Creators:
Alamgir, M1, Author              
von Luxburg, U1, Author              
Affiliations:
1Dept. Empirical Inference, Max Planck Institute for Intelligent Systems, Max Planck Society, DE, ou_1497647              

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 Abstract: We study the family of p-resistances on graphs for p ≥ 1. This family generalizes the standard resistance distance. We prove that for any fixed graph, for p=1, the p-resistance coincides with the shortest path distance, for p=2 it coincides with the standard resistance distance, and for p → ∞ it converges to the inverse of the minimal s-t-cut in the graph. Secondly, we consider the special case of random geometric graphs (such as k-nearest neighbor graphs) when the number n of vertices in the graph tends to infinity. We prove that an interesting phase-transition takes place. There exist two critical thresholds p^* and p^** such that if p < p^*, then the p-resistance depends on meaningful global properties of the graph, whereas if p > p^**, it only depends on trivial local quantities and does not convey any useful information. We can explicitly compute the critical values: p^* = 1 + 1/(d-1) and p^** = 1 + 1/(d-2) where d is the dimension of the underlying space (we believe that the fact that there is a small gap between p^* and p^** is an artifact of our proofs. We also relate our findings to Laplacian regularization and suggest to use q-Laplacians as regularizers, where q satisfies 1/p^* + 1/q = 1.

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 Dates: 2012-01
 Publication Status: Published in print
 Pages: -
 Publishing info: -
 Table of Contents: -
 Rev. Type: -
 Identifiers: BibTex Citekey: Alamgirv2011
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Title: Twenty-Fifth Annual Conference on Neural Information Processing Systems (NIPS 2011)
Place of Event: Granada, Spain
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Title: Advances in Neural Information Processing Systems 24
Source Genre: Proceedings
 Creator(s):
Shawe-Taylor, J, Editor
Zemel, RS, Editor
Bartlett, P, Editor
Pereira, F, Editor
Weinberger, KQ, Editor
Affiliations:
-
Publ. Info: Red Hook, NY, USA : Curran
Pages: - Volume / Issue: - Sequence Number: - Start / End Page: 379 - 387 Identifier: ISBN: 978-1-618-39599-3