The maximum likelihood threshold of a graph

Gross, Elizabeth; Sullivant, Seth

doi:10.3150/16-bej881

Cited by 17 publications

(32 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(1) The generic completion rank of G is at most 2 if and only if G is a looped forest. Equality is attained if and only if G has at least one non-loop edge [6,Theorem 2.5].…”

Section: 1mentioning

confidence: 99%

See 1 more Smart Citation

Typical and generic ranks in matrix completion

Bernstein¹,

Blekherman²,

Sinn³

2020

Linear Algebra and its Applications

View full text Add to dashboard Cite

We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if complex entries are allowed. When the entries are required to be real, this is no longer the case and the possible minimum ranks are called typical ranks. We give a combinatorial description of the patterns of specified entires of n × n symmetric matrices that have n as a typical rank. Moreover, we describe exactly when such a generic partial matrix is minimally completable to rank n. We also characterize the typical ranks for patterns of entries with low maximal typical rank.

show abstract

“…(1) The generic completion rank of G is at most 2 if and only if G is a looped forest. Equality is attained if and only if G has at least one non-loop edge [6,Theorem 2.5].…”

Section: 1mentioning

confidence: 99%

“…In particular, we do not even know of a characterization of the semisimple graphs with generic completion rank two. We therefore also pose the following question, whose answer is known for looped graphs [6] and bipartite graphs [2]. The following proposition gives yet another case study for the disjoint union of graphs.…”

Section: Open Problemsmentioning

confidence: 99%

Typical and generic ranks in matrix completion

Bernstein¹,

Blekherman²,

Sinn³

2020

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…In addition to dependent sets we also have the specific polynomials witnessing the dependencies. This aspect of algebraic matroids has been understood for some time, going back to Dress and Lovász in [10], actually exploiting them in applications seems to be newer (see [15,18,19]).…”

Section: Introductionmentioning

confidence: 99%

Algebraic Matroids in Action

Rosen

Sidman

Theran

2020

The American Mathematical Monthly

View full text Add to dashboard Cite

In recent years, various notions of algebraic independence have emerged as a central and unifying theme in a number of areas of applied mathematics, including algebraic statistics and the rigidity theory of bar-and-joint frameworks. In each of these settings the fundamental problem is to determine the extent to which certain unknowns depend algebraically on given data. This has, in turn, led to a resurgence of interest in algebraic matroids, which are the combinatorial formalism for algebraic (in)dependence. We give a self-contained introduction to algebraic matroids together with examples highlighting their potential application.

show abstract

“…Therefore, it is an important problem to find the minimal number of observations such that the sample covariance matrix has a maximum likelihood estimator with probability one in terms of the graph G underlying the Gaussian graphical model. This number was studied in [7], [25] and following [16] we call it the maximum likelihood threshold of G, denoted mlt(G). See [23] and [26] for an introduction to Gaussian graphical models.…”

Section: Introductionmentioning

confidence: 99%

Maximum Likelihood Threshold and Generic Completion Rank of Graphs

Blekherman

Sinn

2018

Discrete Comput Geom

View full text Add to dashboard Cite

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]

show abstract

The maximum likelihood threshold of a graph

Cited by 17 publications

References 23 publications

Typical and generic ranks in matrix completion

Typical and generic ranks in matrix completion

Algebraic Matroids in Action

Maximum Likelihood Threshold and Generic Completion Rank of Graphs

Contact Info

Product

Resources

About