GBGiovanni Barbarino

Giovanni Barbarino

University of Mons
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Giovanni Barbarino has a total of 25 co-authors. Here are the top co-authors by number of shared publications:

Latest Publications

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    Identifiability of Nonnegative Tucker Decompositions—Part I: Theory

    by Subhayan Saha, Giovanni Barbarino, Nicolas Gillis

    Tensor decompositions have become a central tool in data science, with applications in areas such as data analysis, signal processing, and machine learning. A key property of many tensor decompositions, such as the canonical polyadic decomposition, is identifiability: the factors are unique, up to trivial scaling and permutation ambiguities. This allows one to recover the groundtruth sources that generated the data. The Tucker decomposition (TD) is a central and widely used tensor decomposition model. However, it is in general not identifiable. In this paper, we study the identifiability of the nonnegative TD (nTD). By adapting and extending identifiability results of nonnegative matrix factorization, we provide uniqueness results for nTD. Our results require the nonnegative matrix factors to have some degree of sparsity (namely, satisfy the separability condition, or the sufficiently scattered condition), while the core tensor only needs to have some slices (or linear combinations of them) or unfoldings with full column rank (but does not need to be nonnegative). Under such conditions, we derive several procedures, using either unfoldings or slices of the input tensor, to obtain identifiable nTDs by minimizing the volume of unfoldings or slices of the core tensor.

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    On the Robustness of the Successive Projection Algorithm

    by Giovanni Barbarino, Nicolas Gillis

    The successive projection algorithm (SPA) is a workhorse algorithm to learn the [Formula: see text] vertices of the convex hull of a set of [Formula: see text]-dimensional data points, a.k.a. a latent simplex, which has numerous applications in data science. In this paper, we revisit the robustness to noise of SPA and several of its variants. In particular, when [Formula: see text], we prove the tightness of the existing error bounds for SPA and for two more robust preconditioned variants of SPA. We also provide significantly improved error bounds for SPA, by a factor proportional to the conditioning of the [Formula: see text] vertices, in two special cases: for the first extracted vertex and when [Formula: see text]. We then provide further improvements for the error bounds of a translated version of SPA proposed by Arora et al. [ Proceedings of the International Conference on Machine Learning, 2013, pp. 280–288] in two special cases: for the first two extracted vertices and when [Formula: see text]. Finally, we propose a new more robust variant of SPA that first shifts and lifts the data points in order to minimize the conditioning of the problem. We illustrate our results on synthetic data. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://gitlab.com/ngillis/robustspa . [Formula: see text]

    Cited by 4
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    From asymptotic distribution and vague convergence to uniform convergence, with numerical applications

    by Giovanni Barbarino, Sven-Erik Ekström, Carlo Garoni, David Meadon, Stefano Serra-Capizzano, Paris Vassalos

    Let { Λ n = { λ 1 , n , … , λ d n , n } } n ${\left\{{{\Lambda}}_{n}=\left\{{\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right\}\right\}}_{n}$ be a sequence of finite multisets of real numbers such that d n → ∞ as n → ∞, and let f : Ω ⊂ R d → R $f:{\Omega}\subset {\mathbb{R}}^{d}\to \mathbb{R}$ be a Lebesgue measurable function defined on a domain Ω with 0 < μ d (Ω) < ∞, where μ d is the Lebesgue measure in R d ${\mathbb{R}}^{d}$ . We say that { Λ n } n ${\left\{{{\Lambda}}_{n}\right\}}_{n}$ has an asymptotic distribution described by f , and we write { Λ n } n ∼ f ${\left\{{{\Lambda}}_{n}\right\}}_{n}\sim f$ , if (*) lim n → ∞ 1 d n ∑ i = 1 d n F ( λ i , n ) = 1 μ d ( Ω ) ∫ Ω F ( f ( x ) ) d x ${\mathrm{lim}}_{n\to \infty }\frac{1}{{d}_{n}}{\sum }_{i=1}^{{d}_{n}}F\left({\lambda }_{i,n}\right)=\frac{1}{{\mu }_{d}\left({\Omega}\right)}{\int }_{{\Omega}}F\left(f\left(\boldsymbol{x}\right)\right)\mathrm{d}\boldsymbol{x}$ for every continuous function F with bounded support. If Λ n is the spectrum of a matrix A n , we say that { A n } n ${\left\{{A}_{n}\right\}}_{n}$ has an asymptotic spectral distribution described by f and we write { A n } n ∼ λ f ${\left\{{A}_{n}\right\}}_{n}{ \sim }_{\lambda } f$ . In the case where d = 1, Ω is a bounded interval, Λ n ⊆ f (Ω) for all n , and f satisfies suitable conditions, Bogoya, Böttcher, Grudsky, and Maximenko proved that the asymptotic distribution (*) implies the uniform convergence to 0 of the difference between the properly sorted vector [ λ 1 , n , … , λ d n , n ] $\left[{\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right]$ and the vector of samples [ f ( x 1 , n ) , … , f ( x d n , n ) ] $\left[f\left({x}_{1,n}\right),\dots ,f\left({x}_{{d}_{n},n}\right)\right]$ , i.e., (**) lim n → ∞ max i = 1 , … , d n | f ( x i , n ) − λ τ n ( i ) , n | = 0 , ${\mathrm{lim}}_{n\to \infty } {\mathrm{max}}_{i=1,\dots ,{d}_{n}}\vert f\left({x}_{i,n}\right)-{\lambda }_{{\tau }_{n}\left(i\right),n}\vert =0,$ where x 1 , n , … , x d n , n ${x}_{1,n},\dots ,{x}_{{d}_{n},n}$ is a uniform grid in Ω and τ n is the sorting permutation. We extend this result to the case where d ⩾ 1 and Ω is a Peano–Jordan measurable set (i.e., a bounded set with μ d (∂Ω) = 0). We also formulate and prove a uniform convergence result analogous to (**) in the more general case where the function f takes values in the space of k × k matrices. Our derivations are based on the concept of monotone rearrangement (quantile function) as well as on matrix analysis arguments stemming from the theory of generalized locally Toeplitz sequences and the observation that any finite multiset of numbers Λ n = { λ 1 , n , … , λ d n , n } ${{\Lambda}}_{n}=\left\{{\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right\}$ can always be interpreted as the spectrum of a matrix A n = diag ( λ 1 , n , … , λ d n , n ) ${A}_{n}=\mathrm{diag}\left({\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right)$ . The theoretical results are illustrated through numerical experiments, and a reinterpretation of them in terms of vague convergence of probability measures is hinted.

    Cited by 7
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    Spectral properties of flipped Toeplitz matrices and computational applications

    by Giovanni Barbarino, Sven-Erik Ekström, Carlo Garoni, David Meadon, Stefano Serra-Capizzano, Paris Vassalos

    Cited by 1
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    GLT sequences and normal matrices

    by Giovanni Barbarino, Carlo Garoni

    The theory of generalized locally Toeplitz (GLT) sequences is an apparatus for computing the asymptotic spectral distribution of matrices $A_n$ arising from numerical discretizations of differential equations. Indeed, when the mesh fineness parameter $n$ tends to infinity, these matrices $A_n$ give rise to a sequence $\{A_n\}_n$, which often turns out to be a GLT sequence. In this paper, we extend the theory of GLT sequences in several directions: we show that every GLT sequence enjoys a normal form, we identify the spectral symbol of every GLT sequence formed by normal matrices, and we prove that, for every GLT sequence $\{A_n\}_n$ formed by normal matrices and every continuous function $f:\mathbb C\to\mathbb C$, the sequence $\{f(A_n)\}_n$ is again a GLT sequence whose spectral symbol is $f(\kappa)$, where $\kappa$ is the spectral symbol of $\{A_n\}_n$. In addition, using the theory of GLT sequences, we prove a spectral distribution result for perturbed normal matrices.

    Cited by 1
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    Matrix-less spectral approximation for large structured matrices

    by Giovanni Barbarino, Melker Claesson, Sven-Erik Ekström, Carlo Garoni, David Meadon, Hendrik Speleers

    Sequences of structured matrices of increasing size, such as generalized locally Toeplitz sequences, arise in many scientific applications and especially in the numerical discretization of linear differential problems. We assume that the eigenvalues of a matrix $$X_n$$ X n , belonging to a sequence of such kind, are given by a regular expansion. Under this working hypothesis, we propose a method for computing approximations of the eigenvalues of $$X_n$$ X n for large n and we provide a theoretical analysis of its convergence. The method is called matrix-less because it does not operate on the matrix $$X_n$$ X n but on a few similar matrices of smaller size combined with an interpolation-extrapolation strategy. The working hypothesis and the performance of the proposed eigenvalue approximation method are benchmarked on several numerical examples, with a special attention to matrices arising from the discretization of variable-coefficient differential problems.

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    Dual Simplex Volume Maximization for Simplex-Structured Matrix Factorization

    by Maryam Abdolali, Giovanni Barbarino, Nicolas Gillis

    Simplex-structured matrix factorization (SSMF), closely related to nonnegative matrix factorization, is a fundamental interpretable data analysis model and has applications in hyperspectral unmixing and topic modeling. To obtain identifiable solutions, a standard approach is to find minimum-volume solutions. By taking advantage of the duality/polarity concept for polytopes, we convert minimum-volume SSMF in the primal space to a maximum-volume problem in the dual space. We first prove the identifiability of this maximum-volume dual problem. Then, we use this dual formulation to provide a novel optimization approach which bridges the gap between two existing families of algorithms for SSMF, namely volume minimization and facet identification. Numerical experiments show that the proposed approach performs favorably compared to the state-of-the-art SSMF algorithms.

    Cited by 8
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    Top Cited Papers

    On Properties and Structure of the Analytic Singular Value Decomposition

    by Stephan Weiss, Ian K. Proudler, Giovanni Barbarino, Jennifer Pestana, John G. McWhirter

    We investigate the singular value decomposition (SVD) of a rectangular matrix $\boldsymbol{\mathit{A}}(z)$ of functions that are analytic on an annulus that includes at least the unit circle. Such matrices occur, e.g., as matrices of transfer functions representing broadband multiple-input multiple-output systems. Our analysis is based on findings for the analytic SVD applicable to continuous time systems, and on the analytic eigenvalue decomposition. Using these, we establish two potentially overlapping cases where analyticity of the SVD factors is denied. Firstly, from a structural point of view, multiplexed systems require oversampling by the multiplexing factor in order to admit an analytic solution. Secondly, from an algebraic perspective, we state under which condition spectral zeros of any singular value require additional oversampling by a factor of two if an analytic solution is to be found. In all other cases, an analytic matrix admits an analytic SVD, whereby the singular values are unique up to a permutation, and the left- and right-singular vectors are coupled through a joint ambiguity w.r.t. an arbitrary allpass function. We demonstrate how some state-of-the-art polynomial matrix decomposition algorithms approximate this solution, motivating the need for dedicated algorithms.

    Cited by 42
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    Dual Simplex Volume Maximization for Simplex-Structured Matrix Factorization

    by Maryam Abdolali, Giovanni Barbarino, Nicolas Gillis

    Simplex-structured matrix factorization (SSMF) is a generalization of nonnegative matrix factorization, a fundamental interpretable data analysis model, and has applications in hyperspectral unmixing and topic modeling. To obtain identifiable solutions, a standard approach is to find minimum-volume solutions. By taking advantage of the duality/polarity concept for polytopes, we convert minimum-volume SSMF in the primal space to a maximum-volume problem in the dual space. We first prove the identifiability of this maximum-volume dual problem. Then, we use this dual formulation to provide a novel optimization approach which bridges the gap between two existing families of algorithms for SSMF, namely volume minimization and facet identification. Numerical experiments show that the proposed approach performs favorably compared to the state-of-the-art SSMF algorithms.

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    On the periodicity of singular vectors and the holomorphic block-circulant SVD on the unit circumference

    by Giovanni Barbarino

    Cited by 6
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