2,856 research outputs found
A multi-class approach for ranking graph nodes: models and experiments with incomplete data
After the phenomenal success of the PageRank algorithm, many researchers have
extended the PageRank approach to ranking graphs with richer structures beside
the simple linkage structure. In some scenarios we have to deal with
multi-parameters data where each node has additional features and there are
relationships between such features.
This paper stems from the need of a systematic approach when dealing with
multi-parameter data. We propose models and ranking algorithms which can be
used with little adjustments for a large variety of networks (bibliographic
data, patent data, twitter and social data, healthcare data). In this paper we
focus on several aspects which have not been addressed in the literature: (1)
we propose different models for ranking multi-parameters data and a class of
numerical algorithms for efficiently computing the ranking score of such
models, (2) by analyzing the stability and convergence properties of the
numerical schemes we tune a fast and stable technique for the ranking problem,
(3) we consider the issue of the robustness of our models when data are
incomplete. The comparison of the rank on the incomplete data with the rank on
the full structure shows that our models compute consistent rankings whose
correlation is up to 60% when just 10% of the links of the attributes are
maintained suggesting the suitability of our model also when the data are
incomplete
On wild extensions of a p-adic field
In this paper we consider the problem of classifying the isomorphism classes
of extensions of degree pk of a p-adic field, restricting to the case of
extensions without intermediate fields. We establish a correspondence between
the isomorphism classes of these extensions and some Kummer extensions of a
suitable field F containing K. We then describe such classes in terms of the
representations of Gal(F/K). Finally, for k = 2 and for each possible Galois
group G, we count the number of isomorphism classes of the extensions whose
normal closure has a Galois group isomorphic to G. As a byproduct, we get the
total number of isomorphism classes
Block Tridiagonal Reduction of Perturbed Normal and Rank Structured Matrices
It is well known that if a matrix solves the
matrix equation , where is a linear bivariate polynomial,
then is normal; and can be simultaneously reduced in a finite
number of operations to tridiagonal form by a unitary congruence and, moreover,
the spectrum of is located on a straight line in the complex plane. In this
paper we present some generalizations of these properties for almost normal
matrices which satisfy certain quadratic matrix equations arising in the study
of structured eigenvalue problems for perturbed Hermitian and unitary matrices.Comment: 13 pages, 3 figure
Compression of unitary rank--structured matrices to CMV-like shape with an application to polynomial rootfinding
This paper is concerned with the reduction of a unitary matrix U to CMV-like
shape. A Lanczos--type algorithm is presented which carries out the reduction
by computing the block tridiagonal form of the Hermitian part of U, i.e., of
the matrix U+U^H. By elaborating on the Lanczos approach we also propose an
alternative algorithm using elementary matrices which is numerically stable. If
U is rank--structured then the same property holds for its Hermitian part and,
therefore, the block tridiagonalization process can be performed using the
rank--structured matrix technology with reduced complexity. Our interest in the
CMV-like reduction is motivated by the unitary and almost unitary eigenvalue
problem. In this respect, finally, we discuss the application of the CMV-like
reduction for the design of fast companion eigensolvers based on the customary
QR iteration
A CMV--based eigensolver for companion matrices
In this paper we present a novel matrix method for polynomial rootfinding. By
exploiting the properties of the QR eigenvalue algorithm applied to a suitable
CMV-like form of a companion matrix we design a fast and computationally simple
structured QR iteration.Comment: 14 pages, 4 figure
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Estimating an Eigenvector by the Power Method with a Random Start
This paper addresses the problem of approximating an eigenvector belonging to the largest eigenvalue of a symmetric positive definite matrix by the power method. We assume that the starting vector is randomly chosen with uniform distribution over the unit sphere. This paper provides lower and upper as well as asymptotic bounds on the randomized error in the Lp sense, p ∈[1,+ ∞]. We prove that it is impossible to achieve sharp bounds that are independent of the ratio between the two largest eigenvalues. This should be contrasted to the problem of approximating the largest eigenvalue, for which Kuczyński and Woźniakowski [SIAM J. Matrix Anal. Appl., 13 (1992), pp. 1094--1122] proved that it is possible to bound the randomized error at the kth step with a quantity that depends only on k and on the size of the matrix. We prove that the rate of convergence depends on the ratio of the two largest eigenvalues, on their multiplicities, and on the particular norm. The rate of convergence is at most linear in the ratio of the two largest eigenvalues
Counting Fiedler pencils with repetitions
We introduce a new notation based on diagrams to deal with Fiedler pencils with repetitions (FPR), and use it to solve several counting problems. In particular, we give explicit recurrences to count the number of FPRs of a given degree d, the number of symmetric, palindromic and antipalindromic ones (where the latter two structures are intended in the sense of [5]). We relate these structures to the presence of symmetries in the associated diagrams
A QR based approach for the nonlinear eigenvalue problem
We describe a fast and numerically robust approach based on the structured QR eigenvalue algorithm for computing approximations of the eigenvalues of a holomorphic matrix-valued function inside the unit circle. Numerical experiments confirm the effectiveness of the proposed method
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