Related papers: Sparse Matrix Decompositions and Graph Characteriz…
In this paper, we prove that if the matrix of the linear system is symetric, the Cholesky decomposition can be obtained from the Gauss elimination method without pivoting, without proving that the matrix of the system is positive definite.
Graphs are naturally sparse objects that are used to study many problems involving networks, for example, distributed learning and graph signal processing. In some cases, the graph is not given, but must be learned from the problem and…
In the design of decentralized networked systems, it is useful to know whether a given network topology can sustain stable dynamics. We consider a basic version of this problem here: given a vector space of sparse real matrices, does it…
We study the spectral properties of sparse random graphs with different topologies and type of interactions, and their implications on the stability of complex systems, with particular attention to ecosystems. Specifically, we focus on the…
Exposed positive maps in matrix algebras define a dense subset of extremal maps. We provide a sufficient condition for a positive map to be exposed. This is an analog of a spanning property which guaranties that a positive map is optimal.…
We consider "pressing sequences", a certain kind of transformation of graphs with loops into empty graphs, motivated by an application in phylogenetics. In particular, we address the question of when a graph has precisely one such pressing…
Structure learning methods for covariance and concentration graphs are often validated on synthetic models, usually obtained by randomly generating: (i) an undirected graph, and (ii) a compatible symmetric positive definite (SPD) matrix. In…
We study a weaker formulation of the nullspace property which guarantees recovery of sparse signals from linear measurements by l_1 minimization. We require this condition to hold only with high probability, given a distribution on the…
This paper considers the problem of recovering an unknown sparse p\times p matrix X from an m\times m matrix Y=AXB^T, where A and B are known m \times p matrices with m << p. The main result shows that there exist constructions of the…
The formation trajectory planning using complete graphs to model collaborative constraints becomes computationally intractable as the number of drones increases due to the curse of dimensionality. To tackle this issue, this paper presents a…
Many tools from the field of graph signal processing exploit knowledge of the underlying graph's structure (e.g., as encoded in the Laplacian matrix) to process signals on the graph. Therefore, in the case when no graph is available, graph…
This paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions. Here, sparsistency refers to the property that all…
We consider the sparse inverse covariance regularization problem or graphical lasso with regularization parameter $\rho$. Suppose the co- variance graph formed by thresholding the entries of the sample covariance matrix at $\rho$ is…
The field of compressed sensing has become a major tool in high-dimensional analysis, with the realization that vectors can be recovered from relatively very few linear measurements as long as the vectors lie in a low-dimensional structure,…
How might one "reduce" a graph? That is, generate a smaller graph that preserves the global structure at the expense of discarding local details? There has been extensive work on both graph sparsification (removing edges) and graph…
A concentration graph associated with a random vector is an undirected graph where each vertex corresponds to one random variable in the vector. The absence of an edge between any pair of vertices (or variables) is equivalent to full…
Sparse graphs and their associated matroids play an important role in rigidity theory, where they capture the combinatorics of generically rigid structures. We define a new family called {\bf graded sparse graphs}, arising from generically…
Graph classes of bounded tree rank were introduced recently in the context of the model checking problem for first-order logic of graphs. These graph classes are a common generalization of graph classes of bounded degree and bounded…
We investigate connections between the symmetries (automorphisms) of a graph and its spectral properties. Whenever a graph has a symmetry, i.e. a nontrivial automorphism $\phi$, it is possible to use $\phi$ to decompose any matrix…
Decomposition of large matrix inequalities for matrices with chordal sparsity graph has been recently used by Kojima et al.\ \cite{kim2011exploiting} to reduce problem size of large scale semidefinite optimization (SDO) problems and thus…