Maximum Likelihood Threshold and Generic Completion Rank of Graphs

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Abstract

The minimum number of observations such that the maximum likelihood estimator in a Gaussian graphical model exists with probability one is called the maximum likelihood threshold of the underlying graph G. The natural algebraic relaxation is the generic completion rank introduced by Uhler. We show that the maximum likelihood threshold and the generic completion rank behave in the same way under clique sums, which gives us large families of graphs on which these invariants coincide. On the other hand, we determine both invariants for complete bipartite graphs \(K_{m,n}\) and show that for some choices of m and n the two parameters may be quite far apart. In particular, this gives the first examples of graphs on which the maximum likelihood threshold and the generic completion rank do not agree.

Keywords

Symmetric matrix completion Generic rank Maximum likelihood estimation Graphical models Rigidity Stresses 

Mathematics Subject Classification

14P05 14P25 62-09 52C25 

Notes

Acknowledgements

We thank the referees for their comments. We also thank Seth Sullivant for helpful comments on an earlier version. The authors were partially supported by NSF Grant DMS-1352073.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsGeorgia TechAtlantaUSA
  2. 2.Department of Mathematics and Computer ScienceFreie Universität BerlinBerlinGermany

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