7th Workshop on Matrix Equations and Tensor Techniques

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A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem

The canonical tensor rank approximation problem (TAP) consists of approximating a real-valued tensor by one of low canonical rank, which is a challenging non-linear, non-convex, constrained optimization problem, where the constraint set forms a non-smooth semi-algebraic set. We introduce a Riemannian Gauss-Newton method with trust region for solving small-scale, dense TAPs. The novelty of our approach is threefold. First, we parametrize the constraint set as the Cartesian product of Segre manifolds, hereby formulating the TAP as a Riemannian optimization problem, and we argue why this parametrization is among the theoretically best possible. Second, an original ST-HOSVD-based retraction operator is proposed. Third, we introduce a hot restart mechanism that efficiently detects when the optimization process is tending to an ill-conditioned tensor rank decomposition and which often yields a quick escape path from such spurious decompositions. Numerical experiments show improvements of up to three orders of magnitude in terms of the expected time to compute a successful solution over existing state-of-the-art methods

Rekompresija Hadamardovog produkta tenzora u Tuckerovom formatu

In the last decade low-rank tensors decompositions have been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. Since tensors can be given only as the solution of some algebraic equation, it is important to develop solvers working within the compressed storage scheme. That is what this thesis is concerned with, focusing on Tucker format, one of the most commonly used low-rank representation of tensors, and Hadamard product, which features prominently in tensor-based algorithms in scientific computing and data analysis. Fast algorithms are attained by combining iterative methods, such as Lanczos method and randomized algorithms, with fast matrix-vector products that exploit the structure of Hadamard products. Algorithms are implemented in programming language Julia and a new Julia library for tensors in Tucker format is presented.Posljednjih godina tenzorske dekompozicije malog ranga postaju bitan alat u znanstvenom računanju kod rješavanja problema velikih dimenzija linearne i multilinearne algebre, koje ne možemo riješiti klasičnim tehnikama. S obzirom na to da tenzori mogu biti zadani kao rješenja neke algebarske jednadžbe, izuzetno je važno razviti algoritme koji rade direktno s komprimiranim tenzorskim formatima. U ovoj radnji fokusiramo se na Tuckerov format, jednu od najčešćee korištenih reprezentacija malog ranga, i Hadamardov produkt, koji ima veliku ulogu u tenzorskim algoritmima za znanstveno računanje i obradu podataka. Brze algoritme dobili smo kombinirajući iterativne metode, poput Lanczosove metode i randomiziranih algoritama, s brzim matrično-vektorskim množenjem koje se temelji na posebnoj strukturi Hadamardovog produkta. Algoritmi su implementirani u novu Julia biblioteku

Rekompresija Hadamardovog produkta tenzora u Tuckerovom formatu

In the last decade low-rank tensors decompositions have been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. Since tensors can be given only as the solution of some algebraic equation, it is important to develop solvers working within the compressed storage scheme. That is what this thesis is concerned with, focusing on Tucker format, one of the most commonly used low-rank representation of tensors, and Hadamard product, which features prominently in tensor-based algorithms in scientific computing and data analysis. Fast algorithms are attained by combining iterative methods, such as Lanczos method and randomized algorithms, with fast matrix-vector products that exploit the structure of Hadamard products. Algorithms are implemented in programming language Julia and a new Julia library for tensors in Tucker format is presented.Posljednjih godina tenzorske dekompozicije malog ranga postaju bitan alat u znanstvenom računanju kod rješavanja problema velikih dimenzija linearne i multilinearne algebre, koje ne možemo riješiti klasičnim tehnikama. S obzirom na to da tenzori mogu biti zadani kao rješenja neke algebarske jednadžbe, izuzetno je važno razviti algoritme koji rade direktno s komprimiranim tenzorskim formatima. U ovoj radnji fokusiramo se na Tuckerov format, jednu od najčešćee korištenih reprezentacija malog ranga, i Hadamardov produkt, koji ima veliku ulogu u tenzorskim algoritmima za znanstveno računanje i obradu podataka. Brze algoritme dobili smo kombinirajući iterativne metode, poput Lanczosove metode i randomiziranih algoritama, s brzim matrično-vektorskim množenjem koje se temelji na posebnoj strukturi Hadamardovog produkta. Algoritmi su implementirani u novu Julia biblioteku

Rekompresija Hadamardovog produkta tenzora u Tuckerovom formatu

In the last decade low-rank tensors decompositions have been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. Since tensors can be given only as the solution of some algebraic equation, it is important to develop solvers working within the compressed storage scheme. That is what this thesis is concerned with, focusing on Tucker format, one of the most commonly used low-rank representation of tensors, and Hadamard product, which features prominently in tensor-based algorithms in scientific computing and data analysis. Fast algorithms are attained by combining iterative methods, such as Lanczos method and randomized algorithms, with fast matrix-vector products that exploit the structure of Hadamard products. Algorithms are implemented in programming language Julia and a new Julia library for tensors in Tucker format is presented.Posljednjih godina tenzorske dekompozicije malog ranga postaju bitan alat u znanstvenom računanju kod rješavanja problema velikih dimenzija linearne i multilinearne algebre, koje ne možemo riješiti klasičnim tehnikama. S obzirom na to da tenzori mogu biti zadani kao rješenja neke algebarske jednadžbe, izuzetno je važno razviti algoritme koji rade direktno s komprimiranim tenzorskim formatima. U ovoj radnji fokusiramo se na Tuckerov format, jednu od najčešćee korištenih reprezentacija malog ranga, i Hadamardov produkt, koji ima veliku ulogu u tenzorskim algoritmima za znanstveno računanje i obradu podataka. Brze algoritme dobili smo kombinirajući iterativne metode, poput Lanczosove metode i randomiziranih algoritama, s brzim matrično-vektorskim množenjem koje se temelji na posebnoj strukturi Hadamardovog produkta. Algoritmi su implementirani u novu Julia biblioteku

7th Workshop on Matrix Equations and Tensor Techniques

open6siPrefazione alla special issueopenBenner, Peter; Faßbender, Heike; Grasedyck, Lars; Kressner, Daniel*; Meini, Beatrice; Simoncini, ValeriaBenner, Peter; Faßbender, Heike; Grasedyck, Lars; Kressner, Daniel*; Meini, Beatrice; Simoncini, Valeri

7th Workshop on Matrix Equations and Tensor Techniques

Prefazione alla special issu