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Computing maximum likelihood thresholds using graph rigidity

Daniel Irving Bernstein, Sean Dewar, Steven J. Gortler, Anthony Nixon, Meera Sitharam and Louis Theran

Vol. 14 (2023), No. 2, 287–305
Abstract

The maximum likelihood threshold (MLT) of a graph G is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. We recently proved a new characterization of the MLT in terms of rigidity-theoretic properties of G. This characterization was then used to give new combinatorial lower bounds on the MLT of any graph. We continue this line of research by exploiting combinatorial rigidity results to compute the MLT precisely for several families of graphs. These include graphs with at most nine vertices, graphs with at most 24 edges, every graph sufficiently close to a complete graph and graphs with bounded degrees.

Keywords
maximum likelihood threshold, Gaussian graphical model, combinatorial rigidity, generic completion rank, graph rigidity
Mathematical Subject Classification
Primary: 52C25, 62H12
Milestones
Received: 20 October 2022
Revised: 21 June 2023
Accepted: 30 June 2023
Published: 16 May 2024
Authors
Daniel Irving Bernstein
Department of Mathematics
Tulane University
New Orleans, LA
United States
Sean Dewar
School of Mathematics
University of Bristol
Bristol
United Kingdom
Steven J. Gortler
Department of Computer Science
Harvard University
Cambridge, MA
United States
Anthony Nixon
Department of Mathematics and Statistics
Lancaster University
Lancaster
United Kingdom
Meera Sitharam
Department of Computer and Information Science and Engineering
University of Florida
Gainesville, FL
United States
Louis Theran
School of Mathematics and Statistics
University of St Andrews
St Andrews
United Kingdom