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Lectio Luca Trevisan, "A Theory of Spectral Clustering"
by Federico Poloni 17 Nov '19

17 Nov '19

Segnalo la presenza di questo colloquio, che non fa parte della nostra serie di seminari ma ha un tema che potrebbe essere di interesse per gli iscritti alla lista. È una "lectio magistralis" che fa parte delle celebrazioni per i 50 anni del corso di laurea in informatica. Si terrà nell'aula magna del polo Pontecorvo (edificio E) lunedì 18 novembre alle 15:30. Cordialmente, -federico -------- Forwarded Message -------- Subject: [Dipartimento.di] Lectio Luca Trevisan Date: Fri, 15 Nov 2019 09:55:55 +0100 From: Gian Luigi Ferrari <gian-luigi.ferrari(a)unipi.it> To: CDD <cdd(a)di.unipi.it>, dipartimento(a)di.unipi.it Lo avevo comunicato in consiglio ma mi sono scordato di mandarlo via mail. Rimedio Title: A Theory of Spectral Clustering Speaker: Luca Trevisan Abstract: Spectral clustering algorithms find clusters in a given network by exploiting properties of the eigenvectors of matrices associated with the network. As a first step, one computes a spectral embedding, that is a mapping of nodes to points in a low-dimensional real space; then one uses geometric clustering algorithms such as k-means to cluster the points corresponding to the nodes. Such algorithms work so well that, in certain applications unrelated to network analysis, such as image segmentation, it is useful to associate a network to the data, and then apply spectral clustering to the network. In addition to its application to clustering, spectral embeddings are a valuable tool for dimension-reduction and data visualization. The performance of spectral clustering algorithms has been justified rigorously when applied to networks coming from certain probabilistic generative models. A more recent development, which is the focus of this lecture, is a worst-case analysis of spectral clustering, showing that, for every graph that exhibits a certain cluster structure, such structure can be found by geometric algorithms applied to a spectral embedding. Such results generalize the graph Cheeger’s inequality (a classical result in spectral graph theory), and they have additional applications in computational complexity theory and in pure mathematics. ——— Bio: Luca Trevisan is a professor of Computer Science at Bocconi University. Luca studied at the Sapienza University of Rome, he was a post-doc at MIT and at DIMACS, and he was on the faculty of Columbia University, U.C. Berkeley, and Stanford, before returning to Berkeley in 2014 and, at long last, moving back to Italy in 2019. Luca's research is in theoretical computer science, and it is focused on computational complexity and graph algorithms. Luca received the STOC'97 Danny Lewin (best student paper) award, the 2000 Oberwolfach Prize, and the 2000 Sloan Fellowship. He was an invited speaker at the 2006 International Congress of Mathematicians. He is a recipient of a 2019 ERC Advanced Grant. ______________________________________________ Dipartimento.di mailing list Dipartimento.di(a)listgateway.unipi.it http://listgateway.unipi.it/mailman/listinfo/dipartimento.di

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Seminar on 13/11 (Massimiliano Fasi)
by Leonardo Robol 12 Nov '19

12 Nov '19

Speaker: Massimiliano Fasi Affiliation: Department of Mathematics, The University of Manchester Time: Wednesday, 13/11/2019, 11:00 Place: Aula Seminari, Dipartimento di Matematica Title: Handling the conditioning of extreme-scale matrices In order to assess experimentally the stability of algorithms for the solution of systems of linear equations, it is typically desirable to have a certain degree of control over the condition number of the test matrices being used. If the tests are being performed at scale (e.g., in an HPL benchmark run for a TOP500 submission), it is necessary to ensure that generating the test data will take up only a negligible portion of the overall execution time required to solve the linear system. We develop two new techniques that satisfy these requirements and can be used to efficiently construct extremely large matrices with preassigned 2-norm condition number. Focusing on distributed memory environments, we discuss how these can be implemented in a communication-avoiding fashion. https://www.dm.unipi.it/webnew/it/seminari/handling-conditioning-extreme-sc…

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Colloquio della Classe di Scienze - Alfio Quarteroni
by Leonardo Robol 23 Oct '19

23 Oct '19

Dear all, Alfio Quarteroni will give a presentation at SNS next Wednesday at 15:00, which might be of interest to some of you. The topic is "Taking mathematics to heart"; you can find more information, including venue and abstract, on the SNS website [1]. [1] https://www.sns.it/it/evento/taking-mathematics-to-heart Best wishes, Leonardo Robol.

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Seminari autunnali NumPI
by Leonardo Robol 09 Oct '19

09 Oct '19

Carissimi, prossimamente si terranno alcuni seminari numerici, al Dipartimento di Matematica / Informatica e in Normale, come l'anno scorso. Invito chi di voi fosse potenzialmente interessato a partecipare a compilare il Doodle [1], in modo da poter scegliere un giorno ed un'ora che non coincida con i vostri impegni / lezioni (o almeno, che lo faccia per il minor numero possibile di persone). Per comodità, il Doodle è per date della prossima settimana, ma _non ci sarà_ davvero un seminario la settimana prossima! Probabilmente i primi saranno a fine ottobre / inizio di novembre. Grazie, a presto. [1] https://doodle.com/poll/5p2fm3brdrnu6h4f -- Leonardo Robol <leonardo.robol(a)unipi.it>

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Seminar on 04/07 (Federico Poloni)
by Leonardo Robol 04 Jul '19

04 Jul '19

Speaker: Federico Poloni Affiliation: Department of Computer Science, University of Pisa Time: Thursday, 04/07/2019 Place: Aula Magna, Dipartimento di Matematica Title: Principal pivot transforms, quasidefinite matrices, and matrix equations Principal pivot transforms are maps between matrices that generalize, in some sense, Schur complementation and inversion. They are a somewhat counterintuitive but powerful tool that allows one to reinterpret Gaussian elimination-type algorithms in a different framework. We will show how they can deal successfully with various kinds of matrices in a structure-preserving way; this includes (weakly) quasi-definite matrices, i.e., symmetric matrices with two complementary diagonal blocks of opposite definiteness, as well as Hamiltonian/symplectic matrices. We will then present an algorithm to solve dense algebraic Riccati equations in a factored form, which relies on these transforms as a basic building block.

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Seminar on 25/06 (Michael Overton)
by Leonardo Robol 24 Jun '19

24 Jun '19

Speaker: Michael Overton Affiliation: New York University Time: Wednesday, 25/06/2019, 11:00 Place: Aula Mancini, SNS Title: Crouzeix's conjecture Crouzeix's conjecture is among the most intriguing developments in matrix theory in recent years. Made in 2004 by Michel Crouzeix, it postulates that, for any polynomial p and any matrix A, ||p(A)|| <= 2 max(|p(z)|: z in W(A)), where the norm is the 2-norm and W(A) is the field of values (numerical range) of A, that is the set of points attained by v*Av for some vector v of unit length. Crouzeix proved in 2007 that the inequality above holds if 2 is replaced by 11.08, and recently this was greatly improved by Palencia, replacing 2 by 1+sqrt(2). Furthermore, it is known that the conjecture holds in a number of special cases, including n=2. We use nonsmooth optimization to investigate the conjecture numerically by locally minimizing the “Crouzeix ratio”, defined as the quotient with numerator the right-hand side and denominator the left-hand side of the conjectured inequality. We also present local nonsmooth variational analysis of the Crouzeix ratio at conjectured global minimizers. All our results strongly support the truth of Crouzeix’s conjecture.

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Seminar on 05/06 (Leonardo Robol)
by Leonardo Robol 03 Jun '19

03 Jun '19

Dear all, I am sorry for the delay in this announcement, I was planning to send it before, but it slipped off my mind. Everybody is welcome to attend, as usual. Speaker: Leonardo Robol Affiliation: Department of Mathematics, University of Pisa Time: Wednesday, 05/06/2019 Place: Aula Riunioni, Dipartimento di Matematica Title: Rank structures in matrix equations and matrix functions Low-rank matrices (and tensors) appear often in applications, and this can be exploited in several ways to treat large scale problems that would be otherwise unfeasible. It has been known since a long time that in some applications, such as the solution of the Sylvester equation AX + XB + C = 0, working with low-rank C often leads to a solution X which is full rank, but can be efficiently approximated by a low-rank matrix. This can be used, among other things, in solving PDEs defined by separable operators, such as the 2D Laplace operator on rectangular domains. We briefly review some of these results, and the connection with some rational approximation problems. We show that these structures appear in a multitude of interesting cases. For instance, the same properties can be found when modifying the coefficients of linear matrix equation by performing low-rank updates to their coefficients, and the theory can be extended to more general matrices which have off-diagonal blocks of low-rank. Time permitting, we discuss some recent developments that allow to prove the existence of a low-rank structures in the action of functions of matrices that have a Kronecker sum structure, such as x = f (A⊗I + B^T⊗I)v, for particularly structured vector v. An efficient algorithm for the approximation of the vector x, reshaped in matrix form, is given. This founds applications in extending the ideas used for the Laplace operator and 2D PDEs to more general nonlocal operators, such as the fractional Laplacian.

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Seminar on 11/04 (Fabio Carrara)
by Leonardo Robol 11 Apr '19

11 Apr '19

Speaker: Fabio Carrara Affiliation: ISTI-CNR, Pisa Time: Thursday, 11 April 2019, h. 14:00 Place: Aula Riunioni, Dipartimento di Matematica Title: Neural networks for image understanding: methods, limitations, and new frontiers Vision is natural and easy for living beings, while it is incredibly complex to be replicated in machines. In the last 40 years, a lot of work in Computer Vision has been spent trying to model and implement visual perception manually, with slow improvements over time. However, in the last decade, deep neural networks revolutionized this field by automatically learning complex models for vision from huge amounts of data, thus relieving the practitioners from defining and extracting complex visual features that would require a lot of manual effort to find. In this talk, we will make a journey into neural networks for computer vision focusing on tasks in which they shine, that is image recognition and understanding. We will cover the most successful neural network formulation for visual data, that is the convolutional neural network, from the basics to recent architectures, while pinpointing current limitations in the field. In conclusion, we will discuss novel neural network architectures, such as continuous ODE-defined models, that have been recently proposed and provide new research directions.

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Seminar on 08/04 (Raf Vandebril)
by Leonardo Robol 08 Apr '19

08 Apr '19

Speaker: Raf Vandebril Affiliation: Department of Computer Science, KU Leuven Time: Monday, 08 April 2019, h. 14:00 Place: Aula Seminari, Dipartimento di Matematica Title: Structures, Gramians, and Krylov Krylov methods are typically used for transforming large scale prolems and matrices to matrices of a manageable size. Krylov inspired algorithms are based on the principle: project - solve - lift. In this talk we will focus on the projection step, the orthogonal bases involved, and the structured of the resulting projection. Two bases are needed for the projection: a basis for a search Krylov subspace, say V, and another basis for the constraint Krylov subspace, name it W. For a matrix A, the projected small to medium sized matrix will be W^*AV. There is now a wide variety of possibilities for constructing the matrices W and V. We could go from classical Krylov to extended and rational Krylov subspaces. But we could also attempt to construct subspaces with variants of the matrix A, such as for instance its inverse, its conjugate transpose, or the inverse of its conjugate transpose. In this lecture we will examine some of these possibilities. A general framework is proposed in which we will touch upon the following aspects: biorthogonal (rational) Krylov, CMV factorizations, non-unitary CMV factorizations, quasiseparable Hessenberg matrices, Hankel and Toeplitz Gramians.

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Seminar on 12/04 (James Nagy)
by Leonardo Robol 04 Apr '19

04 Apr '19

Speaker: James Nagy Affiliation: Department of Mathematics, Emory University Time: Friday, 12 April 2019, h. 11:00 Place: Aula Mancini, SNS Title: MATLAB Tools for Large-Scale Linear Inverse Problems Inverse problems arise in a variety of applications: image processing, finance, mathematical biology, and more. Mathematical models for these applications may involve integral equations, partial differential equations, and dynamical systems, and solution schemes are formulated by applying algorithms that incorporate regularization techniques and/or statistical approaches. In most cases these solutions schemes involve the need to solve a large-scale ill-conditioned linear system that is corrupted by noise and other errors. In this talk we describe and demonstrate capabilities of a new MATLAB software package that consists of state-of-the-art iterative methods for solving such systems, which includes approaches that can automatically estimate regularization parameters, stopping iterations, etc., making them very simple to use. Thus, the package allows users to easily incorporate into their own applications (or simply experiment with) different iterative methods and regularization strategies with very little programming effort. On the other hand, sophisticated users can also easily access various options to tune the algorithms for certain applications. Moreover, the package includes several test problems and examples to illustrate how the iterative methods can be used on a variety of large-scale inverse problems. The talk will begin with a brief introduction to inverse problems, discuss considerations that are needed to compute an approximate solution, and describe some details about new efficient hybrid Krylov subspace methods that are implemented in our package. These methods can guide users in automatically choosing regularization parameters, and can be used to enforce various regularization schemes, such as sparsity. We will use imaging examples that arise in medicine and astronomy to illustrate the performance of the methods. This is joint work with Silvia Gazzola (University of Bath) and Per Christian Hansen (Technical University of Denmark).

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