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From Steiner Formulas for Cones to Concentration of Intrinsic Volumes

McCoy, Michael B. and Tropp, Joel A. (2014) From Steiner Formulas for Cones to Concentration of Intrinsic Volumes. Discrete and Computational Geometry, 51 (4). pp. 926-963. ISSN 0179-5376. https://resolver.caltech.edu/CaltechAUTHORS:20140703-103651135

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Abstract

The intrinsic volumes of a convex cone are geometric functionals that return basic structural information about the cone. Recent research has demonstrated that conic intrinsic volumes are valuable for understanding the behavior of random convex optimization problems. This paper develops a systematic technique for studying conic intrinsic volumes using methods from probability. At the heart of this approach is a general Steiner formula for cones. This result converts questions about the intrinsic volumes into questions about the projection of a Gaussian random vector onto the cone, which can then be resolved using tools from Gaussian analysis. The approach leads to new identities and bounds for the intrinsic volumes of a cone, including a near-optimal concentration inequality.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/s00454-014-9595-4DOIArticle
http://link.springer.com/article/10.1007%2Fs00454-014-9595-4PublisherArticle
http://arxiv.org/abs/1308.5265arXivDiscussion Paper
ORCID:
AuthorORCID
Tropp, Joel A.0000-0003-1024-1791
Additional Information:© 2014 Springer Science+Business Media New York. Received: 23 August 2013; Revised: 27 March 2014; Accepted: 1 April 2014. The authors thank Dennis Amelunxen and Martin Lotz for inspiring conversations and for their thoughtful comments on this material. This research was supported by ONR awards N00014-08-1-0883 and N00014-11-1002, AFOSR award FA9550-09-1-0643, and a Sloan Research Fellowship.
Funders:
Funding AgencyGrant Number
Office of Naval Research (ONR)N00014- 08-1-0883
Office of Naval Research (ONR)N00014-11-1002
Air Force Office of Scientific Research (AFOSR)FA9550-09-1-0643
Alfred P. Sloan FoundationUNSPECIFIED
Issue or Number:4
Classification Code:Mathematics Subject Classification: Primary 52A22 · 60D05 · Secondary 52A20
Record Number:CaltechAUTHORS:20140703-103651135
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20140703-103651135
Official Citation:McCoy, M., & Tropp, J. (2014). From Steiner Formulas for Cones to Concentration of Intrinsic Volumes. Discrete & Computational Geometry, 51(4), 926-963. doi: 10.1007/s00454-014-9595-4
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:46808
Collection:CaltechAUTHORS
Deposited By: Jason Perez
Deposited On:07 Jul 2014 23:05
Last Modified:03 Oct 2019 06:46

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