Sharp lower and upper bounds for the Gaussian rank of a graph

Ben-David, Emanuel

doi:10.1016/j.jmva.2015.03.004

Cited by 8 publications

(19 citation statements)

References 11 publications

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“…Again, we state the theorem in terms of the rank of G, using the equivalence established in Theorem 2.5. A variation of Proposition 3.5 with rank replaced by Ben-David's Gaussian rank is proved independently in [2]. Corollary 3.7 was also proven independently in [2] using bounds on the Gaussian rank of a graph.…”

Section: Basic Results On Rank(g)mentioning

confidence: 93%

“…A variation of Proposition 3.5 with rank replaced by Ben-David's Gaussian rank is proved independently in [2]. Corollary 3.7 was also proven independently in [2] using bounds on the Gaussian rank of a graph. While not as powerful as the splitting result from the next section, Corollary 3.7 already implies a number of nice consequences in some simple cases.…”

Section: Basic Results On Rank(g)mentioning

confidence: 93%

“…It should be noted that the rank of a graph is different from the Gaussian rank of a graph as introduced in [2]. The Gaussian rank is also an upper bound on the maximum likelihood threshold of a graph, and it is discussed in more detail in Section 7.…”

Section: Remarkmentioning

confidence: 99%

“…, and V 4 = {4, 8} has each V i an independent set in L (2,4) and each bipartite graph L (2,4) (V i , V j ) without cycles. This implies that rank(G) ≤ 4.…”

Section: Theorem 44 (Splitting Theorem)mentioning

confidence: 99%

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The maximum likelihood threshold of a graph

Gross¹,

Sullivant²

2018

Bernoulli

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The maximum likelihood threshold of a graph is the smallest number of data points that guarantees that maximum likelihood estimates exist almost surely in the Gaussian graphical model associated to the graph. We show that this graph parameter is connected to the theory of combinatorial rigidity. In particular, if the edge set of a graph G is an independent set in the (n − 1)-dimensional generic rigidity matroid, then the maximum likelihood threshold of G is less than or equal to n. This connection allows us to prove many results about the maximum likelihood threshold. We conclude by showing that these methods give exact bounds on the number of observations needed for the score matching estimator to exist with probability one.

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Section: Basic Results On Rank(g)mentioning

confidence: 93%

Section: Basic Results On Rank(g)mentioning

confidence: 93%

Section: Remarkmentioning

confidence: 99%

“…, and V 4 = {4, 8} has each V i an independent set in L (2,4) and each bipartite graph L (2,4) (V i , V j ) without cycles. This implies that rank(G) ≤ 4.…”

Section: Theorem 44 (Splitting Theorem)mentioning

confidence: 99%

See 2 more Smart Citations

The maximum likelihood threshold of a graph

Gross¹,

Sullivant²

2018

Bernoulli

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show abstract

“…The case m = 2 in (c) was known before, see [25,Proposition 4.2]. Another notion related to the generic completion rank, called the Gaussian rank of a graph, was introduced by Ben-David in [3]. The case of complete bipartite graphs shows also that the maximum likelihood threshold can be far from the Gaussian rank of a graph because the Gaussian rank of K m,n is its treewidth, i.e.…”

Section: Introductionmentioning

confidence: 99%

Maximum Likelihood Threshold and Generic Completion Rank of Graphs

Blekherman

Sinn

2018

Discrete Comput Geom

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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.1 arXiv:1703.07849v2 [math.CO]

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On some algorithms for estimation in Gaussian graphical models

Højsgaard,

Lauritzen

2024

Biometrika

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SUMMARY In Gaussian graphical models, the likelihood equations must typically be solved iteratively. We investigate two algorithms: A version of iterative proportional scaling which avoids inversion of large matrices, and an algorithm based on convex duality and operating on the covariance matrix by neighbourhood coordinate descent, corresponding to the graphical lasso with zero penalty. For large, sparse graphs, the iterative proportional scaling algorithm appears feasible and has simple convergence properties. The algorithm based on neighbourhood coordinate descent is extremely fast and less dependent on sparsity, but needs a positive definite starting value to converge. We give an algorithm for finding such a starting value for graphs with low colouring number. As a consequence, we also obtain a simplified proof for existence of the maximum likelihood estimator in such cases.

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Sharp lower and upper bounds for the Gaussian rank of a graph

Cited by 8 publications

References 11 publications

The maximum likelihood threshold of a graph

The maximum likelihood threshold of a graph

Maximum Likelihood Threshold and Generic Completion Rank of Graphs

On some algorithms for estimation in Gaussian graphical models

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