Independence Models for Integer Points of Polytopes.
dc.contributor.author | Shapiro, Austin Warren | en_US |
dc.date.accessioned | 2011-09-15T17:09:32Z | |
dc.date.available | NO_RESTRICTION | en_US |
dc.date.available | 2011-09-15T17:09:32Z | |
dc.date.issued | 2011 | en_US |
dc.date.submitted | 2011 | en_US |
dc.identifier.uri | https://hdl.handle.net/2027.42/86295 | |
dc.description.abstract | The integer points of a high-dimensional polytope P are generally difficult to count or sample uniformly. We consider a class of low-complexity random models for these points which arise from an entropy maximization problem. From these models, by way of "anti-concentration" results for sums of independent random variables, we derive general, efficiently computable upper bounds on the number of integer points of P. We make a detailed study of contingency tables with bounded entries, which are the integer points of a transportation polytope truncated by a cuboid. We provide efficiently computable estimates for the logarithm of the number of m by n tables with specified row and column sums r_1, ..., r_m, c_1, ..., c_n and bounds on the entries. These estimates are asymptotic as m and n go to infinity simultaneously, given that no r_i (resp., c_j) is allowed to exceed a fixed multiple of the average row sum (resp., column sum). As an application, we consider a random, uniformly selected table with entries in {0, 1, ..., kappa} having a given sum. Responding to questions raised by Diaconis and Efron in the context of statistical significance testing, we show that the occurrence of row sums r_1, ..., r_m is positively correlated with the occurrence of column sums c_1, ..., c_n when kappa > 1 and r_1, ..., r_m, c_1, ..., c_n are sufficiently extreme. We give evidence that the opposite is true for near-average values of r_1, ..., r_m, c_1, ..., c_n. | en_US |
dc.language.iso | en_US | en_US |
dc.subject | Polytope | en_US |
dc.subject | Integer Point | en_US |
dc.subject | Lattice Point | en_US |
dc.subject | Littlewood-Offord | en_US |
dc.subject | Maximum Entropy | en_US |
dc.subject | Contingency Table | en_US |
dc.title | Independence Models for Integer Points of Polytopes. | en_US |
dc.type | Thesis | en_US |
dc.description.thesisdegreename | PhD | en_US |
dc.description.thesisdegreediscipline | Mathematics | en_US |
dc.description.thesisdegreegrantor | University of Michigan, Horace H. Rackham School of Graduate Studies | en_US |
dc.contributor.committeemember | Barvinok, Alexandre I. | en_US |
dc.contributor.committeemember | Pettie, Seth | en_US |
dc.contributor.committeemember | Rudelson, Mark | en_US |
dc.contributor.committeemember | Vershynin, Roman | en_US |
dc.subject.hlbsecondlevel | Mathematics | en_US |
dc.subject.hlbtoplevel | Science | en_US |
dc.description.bitstreamurl | http://deepblue.lib.umich.edu/bitstream/2027.42/86295/1/auspex_1.pdf | |
dc.owningcollname | Dissertations and Theses (Ph.D. and Master's) |
Files in this item
Remediation of Harmful Language
The University of Michigan Library aims to describe library materials in a way that respects the people and communities who create, use, and are represented in our collections. Report harmful or offensive language in catalog records, finding aids, or elsewhere in our collections anonymously through our metadata feedback form. More information at Remediation of Harmful Language.
Accessibility
If you are unable to use this file in its current format, please select the Contact Us link and we can modify it to make it more accessible to you.