Low-rank Similarity Measure for Role Model Extraction

Computing meaningful clusters of nodes is crucial to analyze large networks. In this paper, we present a pairwise node similarity measure that allows to extract roles, i.e. group of nodes sharing similar flow patterns within a network. We propose a low rank iterative scheme to approximate the similarity measure for very large networks. Finally, we show that our low rank similarity score successfully extracts the different roles in random graphs and that its performances are similar to the pairwise similarity measure.Comment: 7 pages, 2 columns, 4 figures, conference paper for MTNS201

Structured backward errors for eigenvalues of linear port-Hamiltonian descriptor systems

When computing the eigenstructure of matrix pencils associated with the passivity analysis of perturbed port-Hamiltonian descriptor system using a structured generalized eigenvalue method, one should make sure that the computed spectrum satisfies the symmetries that corresponds to this structure and the underlying physical system. We perform a backward error analysis and show that for matrix pencils associated with port-Hamiltonian descriptor systems and a given computed eigenstructure with the correct symmetry structure there always exists a nearby port-Hamiltonian descriptor system with exactly that eigenstructure. We also derive bounds for how near this system is and show that the stability radius of the system plays a role in that bound

Robustness and perturbations of minimal bases II: The case with given row degrees

This paper studies generic and perturbation properties inside the linear space of $m\times (m+n)$ polynomial matrices whose rows have degrees bounded by a given list $d_1, \ldots, d_m$ of natural numbers, which in the particular case $d_1 = \cdots = d_m = d$ is just the set of $m\times (m+n)$ polynomial matrices with degree at most $d$. Thus, the results in this paper extend to a much more general setting the results recently obtained in [Van Dooren & Dopico, Linear Algebra Appl. (2017), http://dx.doi.org/10.1016/j.laa.2017.05.011] only for polynomial matrices with degree at most $d$. Surprisingly, most of the properties proved in [Van Dooren & Dopico, Linear Algebra Appl. (2017)], as well as their proofs, remain to a large extent unchanged in this general setting of row degrees bounded by a list that can be arbitrarily inhomogeneous provided the well-known Sylvester matrices of polynomial matrices are replaced by the new trimmed Sylvester matrices introduced in this paper. The following results are presented, among many others, in this work: (1) generically the polynomial matrices in the considered set are minimal bases with their row degrees exactly equal to $d_1, \ldots , d_m$, and with right minimal indices differing at most by one and having a sum equal to $\sum_{i=1}^{m} d_i$, and (2), under perturbations, these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly.Comment: arXiv admin note: text overlap with arXiv:1612.0379

A framework for structured linearizations of matrix polynomials in various bases

We present a framework for the construction of linearizations for scalar and matrix polynomials based on dual bases which, in the case of orthogonal polynomials, can be described by the associated recurrence relations. The framework provides an extension of the classical linearization theory for polynomials expressed in non-monomial bases and allows to represent polynomials expressed in product families, that is as a linear combination of elements of the form $\phi_i(\lambda) \psi_j(\lambda)$, where $\{ \phi_i(\lambda) \}$ and $\{ \psi_j(\lambda) \}$ can either be polynomial bases or polynomial families which satisfy some mild assumptions. We show that this general construction can be used for many different purposes. Among them, we show how to linearize sums of polynomials and rational functions expressed in different bases. As an example, this allows to look for intersections of functions interpolated on different nodes without converting them to the same basis. We then provide some constructions for structured linearizations for $\star$-even and $\star$-palindromic matrix polynomials. The extensions of these constructions to $\star$-odd and $\star$-antipalindromic of odd degree is discussed and follows immediately from the previous results

Some numerical challenges in control theory

We discuss a number of novel issues in the interdisciplinary area of numerical linear algebra and control theory. Although we do not claim to be exhaustive we give a number of problems which we believe will play an important role in the near future. These are: sparse matrices, structured matrices, novel matrix decompositions and numerical shortcuts. Each of those is presented in relation to a particular (class of) control problems. These are respectively: large scale control systems, polynomial system models, control of periodic systems, and normalized coprime factorizations in robust control

Maximizing PageRank via outlinks

We analyze linkage strategies for a set I of webpages for which the webmaster wants to maximize the sum of Google's PageRank scores. The webmaster can only choose the hyperlinks starting from the webpages of I and has no control on the hyperlinks from other webpages. We provide an optimal linkage strategy under some reasonable assumptions.Comment: 27 pages, 14 figures, submitted to Linear Algebra App

Descent methods for Nonnegative Matrix Factorization

In this paper, we present several descent methods that can be applied to nonnegative matrix factorization and we analyze a recently developped fast block coordinate method called Rank-one Residue Iteration (RRI). We also give a comparison of these different methods and show that the new block coordinate method has better properties in terms of approximation error and complexity. By interpreting this method as a rank-one approximation of the residue matrix, we prove that it \emph{converges} and also extend it to the nonnegative tensor factorization and introduce some variants of the method by imposing some additional controllable constraints such as: sparsity, discreteness and smoothness.Comment: 47 pages. New convergence proof using damped version of RRI. To appear in Numerical Linear Algebra in Signals, Systems and Control. Accepted. Illustrating Matlab code is included in the source bundl