Recompression of Hadamard products of tensors in Tucker format

doctoral thesis

• Title (croatian): Rekompresija Hadamardovog produkta tenzora u Tuckerovom formatu
• Author: Lana Periša
• Mentor: Ivan Slapničar (mentor)
• Mentor: Daniel Kressner (komentor); MBZ: 283256
• Committee member: Luka Grubišić (predsjednik povjerenstva)
• Committee member: Zvonimir Bujanović (član povjerenstva)
• Committee member: Ninoslav Truhar (član povjerenstva)
• Committee member: Ivan Slapničar (član povjerenstva)
• Committee member: Daniel Kressner (član povjerenstva); MBZ: 283256
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-11-23, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

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.

Abstract (croatian)

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.

• Keywords: tensors; Tucker format; Tucker decomposition; higher-order singular value decomposition; HOSVD; Hadamard products; low-rank approximation
• Keywords (croatian): tenzori; Tuckerov format; Tuckerov rastav; singularna dekompozicija više reda; HOSVD; Hadamardov produkt; aproksimacija malog ranga
• Language: english
• URN:NBN: urn:nbn:hr:217:508763
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: vi, 111 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-03-21 10:02:06