Positive semidefinite matrix completion, universal rigidity and the Strong Arnold Property

Laurent, Monique; Varvitsiotis, Antonios

doi:10.1016/j.laa.2014.03.015

Cited by 28 publications

(40 citation statements)

References 30 publications

(96 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For linear SDP, it reduces to the primal (dual) nondegeneracy of Alizadeh et al [1], see also Chan and Sun [7] for further deep implications in SDP. It has been shown fundamental in many optimization problems, see [40,33,39,34,26]. Our main result is that constraint nondegeneracy holds for the convex problem (11) under a very weak condition and it further ensures that the Newton-CG method is quadratically convergent.…”

Section: The Matrix Optimization Formulationmentioning

confidence: 84%

“…To ensure the existence of γ that satisfies (26), it is enough to prove the linear independence of the vectors {W i u} n i=1 . Assume that there exist…”

Section: The Matrix Optimization Formulationmentioning

confidence: 99%

“…, n. Hence, the vectors {W i u} n i=1 are linearly independent. There is a unique γ satisfying (26). Therefore, γ = U T γ. Γ is well defined and the resulting X defined in (18) satisfies (23).…”

Section: The Matrix Optimization Formulationmentioning

confidence: 99%

See 2 more Smart Citations

Constrained Best Euclidean Distance Embedding on a Sphere: A Matrix Optimization Approach

Bai

Xiu³

2015

SIAM J. Optim.

View full text Add to dashboard Cite

Abstract. The problem of data representation on a sphere of unknown radius arises from various disciplines such as Statistics (spatial data representation), Psychology (constrained multidimensional scaling), and Computer Science (machine learning and pattern recognition). The best representation often needs to minimize a distance function of the data on a sphere as well as to satisfy some Euclidean distance constraints. It is those spherical and Euclidean distance constraints that present an enormous challenge to the existing algorithms. In this paper, we reformulate the problem as an Euclidean distance matrix optimization problem with a low rank constraint. We then propose an iterative algorithm that uses a quadratically convergent Newton-CG method at its each step. We study fundamental issues including constraint nondegeneracy and the nonsingularity of generalized Jacobian that ensure the quadratic convergence of the Newton method. We use some classic examples from the spherical multidimensional scaling to demonstrate the flexibility of the algorithm in incorporating various constraints. We also present an interesting application to the circle fitting problem.Key words. Euclidean distance matrix, Matrix optimization, Lagrangian duality, Spherical multidimensional scaling, Semismooth Newton-CG method.AMS subject classifications. 49M45, 90C25, 90C331. Introduction. In this paper, we are mainly concerned with placing n points {x 1 , . . . , x n } in a best way on a sphere in IR r . The primary information that we use is an incomplete/complete set of pairwise Euclidean distances (often with noises) among the n points. In such a setting, IR r is often a low-dimensional space (e.g., r takes 2 or 3 for data visualization) and is known as the embedding space. The center of the sphere is unknown. For some applications, the center can be put at origin in IR r . Furthermore, the radius of the sphere is also unknown. In our matrix optimization formulation of the problem, we treat both the center and the radius as unknown variables. We develop a fast numerical method for this problem and present a few of interesting applications taken from existing literature.The problem described above has long appeared in the constrained Multi-Dimensional Scaling (MDS) when r ≤ 3, which is mainly for the purpose of data visualization, see [9, Sect. 4.6] and [4, Sect. 10.3] for more details. In particular, it is known as the spherical MDS when r = 3 and the circular MDS when r = 2. Most numerical methods in this part took advantages of r being 2 or 3. For example, two of the earliest circular MDS were by Borg and Lingoes [5] and Lee and Bentler [28], where they introduced a new point x 0 ∈ IR r as the center of the sphere (i.e., circles in their case) and further forced the following constraints to hold:

show abstract

Section: The Matrix Optimization Formulationmentioning

confidence: 84%

“…To ensure the existence of γ that satisfies (26), it is enough to prove the linear independence of the vectors {W i u} n i=1 . Assume that there exist…”

Section: The Matrix Optimization Formulationmentioning

confidence: 99%

See 1 more Smart Citation

Constrained Best Euclidean Distance Embedding on a Sphere: A Matrix Optimization Approach

Bai

Xiu³

2015

SIAM J. Optim.

View full text Add to dashboard Cite

show abstract

“…We will return to this in section 2.3 below. Laurent and Varvitsiotis [14] also worked on the link between rigidity and completability, where they discussed the relation between universal completability and universal rigidity in terms of SDP formulations, again assuming that G contains a loop at every vertex.…”

Section: Previous Workmentioning

confidence: 99%

Combinatorial Conditions for the Unique Completability of Low-Rank Matrices

Jackson

Jordán

Tanigawa

2014

SIAM J. Discrete Math.

View full text Add to dashboard Cite

We consider the problems of completing a low-rank positive semidefinite square matrix M or a low-rank rectangular matrix N from a given subset of their entries. Following the approach initiated by Singer and Cucuringu [20] we study the local and global uniqueness of such completions by analysing the structure of the graphs determined by the positions of the known entries of M or N .We present combinatorial characterizations of local and global (unique) completability for special families of graphs. We characterize local and global completability in all dimensions for cluster graphs, i.e. graphs which can be obtained from disjoint complete graphs by adding a set of independent edges. These results correspond to theorems for body-bar frameworks in rigidity theory. We also provide a characterization of two-dimensional local completability of planar bipartite graphs, which leads to a characterization of two-dimensional local completability in the rectangular matrix model when the underlying bipartite graph is planar. These results are based on new observations that certain graph operations preserve local or global completability, as well as on a further connection between rigidity and completability.We also prove that a rank condition on the completability stress matrix of a graph is a sufficient condition for global completability. This verifies a conjecture of Singer and Cucuringu [20].

show abstract

“…Generic completion rank for symmetric matrices has applications in statistics as a bound for the maximum likelihood threshold of a Gaussian graphical model [4,6,13]. If we restrict to positive semidefinite completions, then maximal typical rank of a pattern (suitably defined) is known as the Gram dimension, and is closely related to Euclidean distance realization problems [9,10]. We note that for the positive semidefinite matrix completion, there are no partial matrices that can be completed only to full rank as any entry of a positive definite matrix may be changed to make the matrix positive semidefinite and drop rank.…”

Section: Introductionmentioning

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

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

Positive semidefinite matrix completion, universal rigidity and the Strong Arnold Property

Cited by 28 publications

References 30 publications

Constrained Best Euclidean Distance Embedding on a Sphere: A Matrix Optimization Approach

Constrained Best Euclidean Distance Embedding on a Sphere: A Matrix Optimization Approach

Combinatorial Conditions for the Unique Completability of Low-Rank Matrices

Typical and generic ranks in matrix completion

Contact Info

Product

Resources

About