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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Maximum likelihood thresholds via graph rigidity
AU - Bernstein, Daniel
AU - Dewar, Sean
AU - Gortler, Steven
AU - Nixon, Anthony
AU - Sitharam, Meera
AU - Theran, Louis
PY - 2024/6/13
Y1 - 2024/6/13
N2 - 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 give a new characterization of the MLT in terms of rigidity-theoretic properties of G and use this characterization to give new combinatorial lower bounds on the MLT of any graph.We use the new lower bounds to give high-probability guarantees on the maximum likelihood thresholds of sparse Erdős–Rényi random graphs in terms of their average density. These examples show that the new lower bounds are within a polylog factor of tight, where, on the same graph families, all known lower bounds are trivial.Based on computational experiments made possible by our methods, we conjecture that the MLT of an Erdős–Rényi random graph is equal to its generic completion rank with high probability. Using structural results on rigid graphs in low dimension, we can prove the conjecture for graphs with MLT at most 4 and describe the threshold probability for the MLT to switch from 3 to 4.We also give a geometric characterization of the MLT of a graph in terms of a new “lifting” problem for frameworks that is interesting in its own right. The lifting perspective yields a new connection between the weak MLT (where the maximum likelihood estimate exists only with positive probability) and the classical Hadwiger–Nelson problem
AB - 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 give a new characterization of the MLT in terms of rigidity-theoretic properties of G and use this characterization to give new combinatorial lower bounds on the MLT of any graph.We use the new lower bounds to give high-probability guarantees on the maximum likelihood thresholds of sparse Erdős–Rényi random graphs in terms of their average density. These examples show that the new lower bounds are within a polylog factor of tight, where, on the same graph families, all known lower bounds are trivial.Based on computational experiments made possible by our methods, we conjecture that the MLT of an Erdős–Rényi random graph is equal to its generic completion rank with high probability. Using structural results on rigid graphs in low dimension, we can prove the conjecture for graphs with MLT at most 4 and describe the threshold probability for the MLT to switch from 3 to 4.We also give a geometric characterization of the MLT of a graph in terms of a new “lifting” problem for frameworks that is interesting in its own right. The lifting perspective yields a new connection between the weak MLT (where the maximum likelihood estimate exists only with positive probability) and the classical Hadwiger–Nelson problem
U2 - 10.1214/23-AAP2039
DO - 10.1214/23-AAP2039
M3 - Journal article
VL - 34
SP - 3288
EP - 3319
JO - Annals of Applied Probability
JF - Annals of Applied Probability
SN - 1050-5164
IS - 3
ER -