Klasifikacija slika tenzorskim metodama

master's thesis

• Title (english): Image classification using tensor methods
• Author: Luka Jandrijević
• Mentor: Zlatko Drmač (mentor)
• Committee member: Tina Bosner (predsjednik povjerenstva)
• Committee member: Matko Ljulj (član povjerenstva)
• Committee member: Tomislav Pejković (član povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu su opisani pojmovi i operacije usko vezane uz multilinearnu algebru i objekte koje nazivamo tenzori. Opisane su osnovne operacije koje provodimo nad tenzorima te opisujemo matričnu dekompoziciju SVD te taj koncept proširujemo na tenzore. Zatim, promatramo još jednu tenzorsku dekompoziciju pod nazivom tensor train dekompozicija koja ne ovisi toliko o redu tenzora koliko HOSVD ovisi. Dekompozicija tenzora pomoću HOSVD-a i tensor train-a su također implementirane u ovom radu kako bi se njihov algoritam što bolje razumio. Oba algoritma zasnivana su na dokazima tvrdnji koje daju konstruktivan način njihove egzistencije. Pomoću tih algoritama možemo tenzore aproksimirati vrlo dobro. Samim time i zbog toga što slike možemo također reprezentirati pomoću tenzora, koristimo ta dva algoritma kako bi aproksimirali slike. Rješavamo problem klasifikacije rukom pisanih znamenaka koristeći ove dvije dekompozicije pri čemu su se tenzori pokazali izuzetno korisnim. Prvi način rješavanja problema klasifikacije znamenaka smo riješili pomoću HOSVD metode. Zatim, isti taj problem rješavamo pomoću tensor train dekompozicije. Oba riješenja pokazala su se efikasnima te su sve implementacije rađene u programskom jeziku Python. Ipak, pomoću HOSVD algoritma smo dobili malo veću preciznost nego s tensor train algoritmom. Također, biramo različite parametre kako bi dostigli različite točnosti. Bitno je vrlo pažljivo birati takve parametre kako bi dostigli što veću preciznost algoritma jer ponekad nije najbolji pristup birati što veće parametre kao što smo to vidjeli na primjeru klasifikacije uz pomoć HOSVD algoritma. Na kraju rada također možemo vidjeti koje znamenke su algoritmi krivo pogodili. Očekivano, neke jednostavnije znamenke imaju veću točnost pogađanja od drugih.

Abstract (english)

This paper describes concepts and operations closely related to multilinear algebra and objects known as tensors. It explains basic operations performed on tensors and extends the concept of matrix decomposition through SVD to tensors. Additionally, it explores another tensor decomposition method called tensor train decomposition, which is less dependent on tensor rank compared to HOSVD. Tensor decompositions using HOSVD and tensor train are implemented in this study to better understand their algorithms. Both algorithms are based on proofs that provide constructive ways of obtaining them. These algorithms are effective for approximating tensors, especially given that images can also be represented as tensors. Therefore, both methods are used to approximate images. The problem of classifying handwritten digits is addressed using these two tensor decompositions, and tensors have proven to be exceptionally useful in this context. The first approach to digit classification is solved using the HOSVD method. Subsequently, the same problem is tackled using tensor train decomposition. Both solutions are efficient, and all implementations are carried out in the Python programming language. However, the HOSVD algorithm yields slightly higher accuracy compared to the tensor train algorithm. Additionally, different parameters are chosen to achieve varying levels of accuracy. Careful parameter selection is essential for achieving high algorithm accuracy, as demonstrated in the digit classification example using the HOSVD algorithm. Finally, the paper presents the misclassified digits, where it is expected that simpler digits have higher accuracy in classification than others.

• Keywords: multilinearna algebra; matrična dekompozicija SVD; HOSVD; Python
• Keywords (english): multilinear algebra; matrix decomposition through SVD; HOSVD; Python
• Language: croatian
• URN:NBN: urn:nbn:hr:217:450303
• Study programme: Title: Computer Science and Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar računarstva i matematike (sveučilišni magistar računarstva i matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-02-01 13:14:56