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Giovanni Barbarino

Publications and source records attributed to Giovanni Barbarino.

31 records · Page 2Linked to original sources
arXiv2022

On role extraction for digraphs via neighbourhood pattern similarity

We analyse the recovery of different roles in a network modelled by a directed graph, based on the so-called Neighbourhood Pattern Similarity approach. Our analysis uses results from random matrix theory to show that when assuming the graph is generated as a particular Stochastic Block Model with Bernoulli probability distributions for the different blocks, then the recovery is asymptotically correct when the graph has a sufficiently large dimension. Under these assumptions there is a sufficient gap between the dominant and dominated eigenvalues of the similarity matrix, which guarantees the asymptotic correct identification of the number of different roles. We also comment on the connections with the literature on Stochastic Block Models, including the case of probabilities of order log(n)/n where n is the graph size. We provide numerical experiments to assess the effectiveness of the method when applied to practical networks of finite size.

math.NA↗
arXiv2022

Conjectures on spectral properties of ALIF algorithm

A new decomposition method for nonstationary signals, named Adaptive Local Iterative Filtering (ALIF), has been recently proposed in the literature. Given its similarity with the Empirical Mode Decomposition (EMD) and its more rigorous mathematical structure, which makes feasible to study its convergence compared to EMD, ALIF has really good potentiality to become a reference method in the analysis of signals containing strong nonstationary components, like chirps, multipaths and whistles, in many applications, like Physics, Engineering, Medicine and Finance, to name a few. In [11], the authors analyzed the spectral properties of the matrices produced by the ALIF method, in order to study its stability. Various results are achieved in that work through the use of Generalized Locally Toeplitz (GLT) sequences theory, a powerful tool originally designed to extract information on the asymptotic behavior of the spectra for PDE discretization matrices. In this manuscript we focus on answering some of the open questions contained in [11], and in doing so, we also develop new theory and results for the GLT sequences.

math.NA↗
arXiv2022

Theoretical results for eigenvalues, singular values, and eigenvectors of (flipped) Toeplitz matrices and related computational proposals

In a series of recent papers the spectral behavior of the matrix sequence $\{Y_nT_n(f)\}$ is studied in the sense of the spectral distribution, where $Y_n$ is the main antidiagonal (or flip matrix) and $T_n(f)$ is the Toeplitz matrix generated by the function $f$, with $f$ being Lebesgue integrable and with real Fourier coefficients. This kind of study is also motivated by computational purposes for the solution of the related large linear systems using the (preconditioned) MINRES algorithm. Here we complement the spectral study with more results holding both asymptotically and for a fixed dimension $n$, and with regard to eigenvalues, singular values, and eigenvectors of $T_n(f), Y_nT_n(f)$ and to several relationships among them: beside fast linear solvers, a further target is the design of ad hoc procedures for the computation of the related spectra via matrix-less algorithms, with a cost being linear in the number of computed eigenvalues. We emphasize that the challenge of the case of non-monotone generating functions is considered in the current work, for which the previous matrix-less algorithms fail. Numerical experiments are reported and commented, with the aim of showing in a visual way the theoretical analysis.

math.NA↗
arXiv2022

The limit empirical spectral distribution of Gaussian monic complex matrix polynomials

We define the empirical spectral distribution (ESD) of a random matrix polynomial with invertible leading coefficient, and we study it for complex $n \times n$ Gaussian monic matrix polynomials of degree $k$. We obtain exact formulae for the almost sure limit of the ESD in two distinct scenarios: (1) $n \rightarrow \infty$ with $k$ constant and (2) $k \rightarrow \infty$ with $n$ constant. The main tool for our approach is the replacement principle by Tao, Vu and Krishnapur. Along the way, we also develop some auxiliary results of potential independent interest: we slightly extend a result by Bürgisser and Cucker on the tail bound for the norm of the pseudoinverse of a non-zero mean matrix, and we obtain several estimates on the singular values of certain structured random matrices.

math.PR↗
arXiv2022

Rectangular GLT Sequences

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of square matrices $A_n$ arising from the discretization of differential problems. Indeed, as the mesh fineness parameter $n$ increases to $\infty$, the sequence $\{A_n\}_n$ often turns out to be a GLT sequence. In this paper, motivated by recent applications, we further enhance the GLT apparatus by developing a full theory of rectangular GLT sequences as an extension of the theory of classical square GLT sequences. We also detail an example of application as an illustration of the potential impact of the theory presented herein.

math.NA↗
arXiv2021

Constructive approach to the monotone rearrangement of functions

We detail a simple procedure (easily convertible to an algorithm) for constructing from quasi-uniform samples of $f$ a sequence of linear spline functions converging to the monotone rearrangement of $f$, in the case where $f$ is an almost everywhere continuous function defined on a bounded set $Ω$ with negligible boundary. Under additional assumptions on $f$ and $Ω$, we prove that the convergence of the sequence is uniform. We also show that the same procedure applies to arbitrary measurable functions too, but with the substantial difference that in this case the procedure has only a theoretical interest and cannot be converted to an algorithm.

math.NA↗
arXiv2021

The limit empirical spectral distribution of complex matrix polynomials

We study the empirical spectral distribution (ESD) for complex n x n matrix polynomials of degree k. We obtain exact formulae for the almost sure limit of the ESD in two distinct scenarios: (1) n -> \infty with k constant and (2) k -> \infty with n bounded by O(k^P) for some P>0. The main tools used are the logarithmic potential of some measure related to the matrix polynomial, and some classical estimates on the singular values of full random matrices with i.i.d. entries.

math.PR↗
arXiv2021

A systematic approach to reduced GLT

This paper concerns the spectral analysis of matrix-sequences that are generated by the discretization and numerical approximation of partial differential equations (PDEs), in case the domain is a generic Peano-Jordan measurable set. It is observed that such matrix-sequences often present a spectral symbol, that is a measurable function describing the asymptotic behaviour of the eigenvalues. When the domain is a hypercube, the analysis can be conducted using the theory of generalized locally Toeplitz (GLT) sequences, but in case of generic domain, a new kind of matrix-sequences and theory has to be formalized. We thus introduce the Reduced GLT sequences and symbols, developing in full detail its theory, and presenting some application to finite differences and finite elements discretization for convection-diffusion-reaction differential equations.

math.NA↗
arXiv2018

Notes on asymptotic eigenvalues distribution on complex circles

With tools of measure theory and symbols of matrix sequences, we explore the results regarding curves on finite fields and Weil Systems. This document wants to draw a bridge between the two areas and link the concepts of distribution of Frobenius roots in the context of Zeta functions on curves, with the spectral distributions already studied in linear algebra.

math.FA↗
arXiv2017

Diagonal Matrix Sequences and their Spectral Symbols

The spectral symbols are useful tools to analyse the eigenvalue distribution when dealing with high dimensional linear systems. Given a matrix sequence with an asymptotic symbol, the last one depends only on the spectra of the individual matrices, seen as a not ordered set. We can then focus only on diagonal sequences and sort the eigenvalues so that they become an approximation of the symbol sampling. We show that this is linked to the concept of diagonal Generalized Locally Toeplitz (GLT) sequences, and in particular we prove that any diagonal sequence with a real valued symbol can be permuted in order to obtain a diagonal GLT sequence with the same symbol.

math.NA↗
arXiv2017

Equivalence between GLT sequences and measurable functions

The theory of Generalized Locally Toeplitz (GLT) sequences of matrices has been developed in order to study the asymptotic behaviour of particular spectral distributions when the dimension of the matrices tends to infinity. A key concepts in this theory are the notion of Approximating Classes of Sequences (a.c.s.), and spectral symbols, that lead to define a metric structure on the space of matrix sequences, and provide a link with the measurable functions. In this document we prove additional results regarding theoretical aspects, such as the completeness of the matrix sequences space with respect to the metric a.c.s., and the identification of the space of GLT sequences with the space of measurable functions.

math.NA↗
arXiv2016

Permutation NMF

Nonnegative Matrix Factorization(NMF) is a common used technique in machine learning to extract features out of data such as text documents and images thanks to its natural clustering properties. In particular, it is popular in image processing since it can decompose several pictures and recognize common parts if they're located in the same position over the photos. This paper's aim is to present a way to add the translation invariance to the classical NMF, that is, the algorithms presented are able to detect common features, even when they're shifted, in different original images.

cs.CV↗