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The Metric Nearness Problem

Brickell, Justin and Dhillon, Inderjit S. and Sra, Suvrit and Tropp, Joel A. (2008) The Metric Nearness Problem. SIAM Journal on Matrix Analysis and Applications, 30 (1). pp. 375-396. ISSN 0895-4798. doi:10.1137/060653391.

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Metric nearness refers to the problem of optimally restoring metric properties to distance measurements that happen to be nonmetric due to measurement errors or otherwise. Metric data can be important in various settings, for example, in clustering, classification, metric-based indexing, query processing, and graph theoretic approximation algorithms. This paper formulates and solves the metric nearness problem: Given a set of pairwise dissimilarities, find a “nearest” set of distances that satisfy the properties of a metric—principally the triangle inequality. For solving this problem, the paper develops efficient triangle fixing algorithms that are based on an iterative projection method. An intriguing aspect of the metric nearness problem is that a special case turns out to be equivalent to the all pairs shortest paths problem. The paper exploits this equivalence and develops a new algorithm for the latter problem using a primal-dual method. Applications to graph clustering are provided as an illustration. We include experiments that demonstrate the computational superiority of triangle fixing over general purpose convex programming software. Finally, we conclude by suggesting various useful extensions and generalizations to metric nearness.

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Tropp, Joel A.0000-0003-1024-1791
Additional Information:Received by the editors March 2, 2006; accepted for publication (in revised form) by R. Nabben September 11, 2007; published electronically April 23, 2008. This research was supported by NSF grant CCF-0431257, NSF Career Award ACI-0093404, and NSF-ITR award IIS-0325116. A preliminary version of this work appeared at NIPS 2004, Vancouver, Canada.
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Subject Keywords:matrix nearness problems, metric, distance matrix, metric nearness, all pairs shortest paths, triangle inequality
Issue or Number:1
Classification Code:AMS: 05C12, 05C85, 54E35, 65Y20, 90C06, 90C08
Record Number:CaltechAUTHORS:20110513-152152857
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:23672
Deposited On:13 May 2011 22:32
Last Modified:09 Nov 2021 16:16

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