164 research outputs found
A new graph parameter related to bounded rank positive semidefinite matrix completions
The Gram dimension \gd(G) of a graph is the smallest integer
such that any partial real symmetric matrix, whose entries are specified on the
diagonal and at the off-diagonal positions corresponding to edges of , can
be completed to a positive semidefinite matrix of rank at most (assuming a
positive semidefinite completion exists). For any fixed the class of graphs
satisfying \gd(G) \le k is minor closed, hence it can characterized by a
finite list of forbidden minors. We show that the only minimal forbidden minor
is for and that there are two minimal forbidden minors:
and for . We also show some close connections to
Euclidean realizations of graphs and to the graph parameter of
\cite{H03}. In particular, our characterization of the graphs with \gd(G)\le
4 implies the forbidden minor characterization of the 3-realizable graphs of
Belk and Connelly \cite{Belk,BC} and of the graphs with of van
der Holst \cite{H03}.Comment: 31 pages, 6 Figures. arXiv admin note: substantial text overlap with
arXiv:1112.596
A new graph parameter related to bounded rank positive semidefinite matrix completions.
The Gram dimension gd(G) of a graph G is the smallest inte- ger k ≥ 1 such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding to edges of G, can be completed to a positive semidefinite matrix of rank at most k (assuming a positive semidefinite completion exists). For any fixed k the class of graphs satisfying gd(G) ≤ k is minor closed, hence it can characterized by a finite list of forbidden minors. We show that the only minimal forbidden minor is Kk+1 for k ≤ 3 and that there are two minimal forbidden minors: K5 and K2,2,2 for k = 4. We also show some close connections to Euclidean realizations of graphs and to the graph parameter ν=(G) of [21]. In particular, our characterization of the graphs with gd(G) ≤ 4 implies the forbidden minor characterization of the 3-realizable graphs of Belk and Connelly [8,9] and of the graphs with ν=(G) ≤ 4 of van der Holst [21]
Forbidden minor characterizations for low-rank optimal solutions to semidefinite programs over the elliptope
We study a new geometric graph parameter \egd(G), defined as the smallest
integer for which any partial symmetric matrix which is completable to
a correlation matrix and whose entries are specified at the positions of the
edges of , can be completed to a matrix in the convex hull of correlation
matrices of \rank at most . This graph parameter is motivated by its
relevance to the problem of finding low rank solutions to semidefinite programs
over the elliptope, and also by its relevance to the bounded rank Grothendieck
constant. Indeed, \egd(G)\le r if and only if the rank- Grothendieck
constant of is equal to 1. We show that the parameter \egd(G) is minor
monotone, we identify several classes of forbidden minors for \egd(G)\le r
and we give the full characterization for the case . We also show an upper
bound for \egd(G) in terms of a new tree-width-like parameter \sla(G),
defined as the smallest for which is a minor of the strong product of a
tree and . We show that, for any 2-connected graph on at
least 6 nodes, \egd(G)\le 2 if and only if \sla(G)\le 2.Comment: 33 pages, 8 Figures. In its second version, the paper has been
modified to accommodate the suggestions of the referees. Furthermore, the
title has been changed since we feel that the new title reflects more
accurately the content and the main results of the pape
Chordal Decomposition in Rank Minimized Semidefinite Programs with Applications to Subspace Clustering
Semidefinite programs (SDPs) often arise in relaxations of some NP-hard
problems, and if the solution of the SDP obeys certain rank constraints, the
relaxation will be tight. Decomposition methods based on chordal sparsity have
already been applied to speed up the solution of sparse SDPs, but methods for
dealing with rank constraints are underdeveloped. This paper leverages a
minimum rank completion result to decompose the rank constraint on a single
large matrix into multiple rank constraints on a set of smaller matrices. The
re-weighted heuristic is used as a proxy for rank, and the specific form of the
heuristic preserves the sparsity pattern between iterations. Implementations of
rank-minimized SDPs through interior-point and first-order algorithms are
discussed. The problem of subspace clustering is used to demonstrate the
computational improvement of the proposed method.Comment: 6 pages, 6 figure
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
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