HBM 698E - Computational Methods for Tensor Decomposition and Low-Rank Approximation

Course Code HBM698E

Course Title Computational Methods for Tensor Decomposition and Low-Rank Approximation

Instructor Assoc. Prof. Dr. Süha Tuna

Department / Level Computational Science & Engineering (PhD)

Course Overview

This advanced PhD level course covers fundamental matrix decompositions, spectral decomposition, tensors and their mathematical properties, tensor folding, matricization, vectorization, tensor products, Kruskal rank, tensor decompositions, Tucker/CP/HOSVD decompositions, Tensor Train decomposition, Tensor Ring decomposition, High Dimensional Model Representation (HDMR), Enhanced Multivariance Products Representation (EMPR), Holistic Multivariance Decomposition (HMD) and applications of tensor decompositions to complex multidimensional problems.

Syllabus & Lecture Topics

Week	Topic & Detailed Content
Week 1	Linearity, Linear Independence, Linear Vector Spaces, Span, and Basis
Week 2	Four Fundamental Subspaces of Matrices, Eigenvalues, and Eigenvectors
Week 3	Spectral Decomposition and Singular Value Decomposition (SVD)
Week 4	Rank-1 Decomposition and Low-Rank Approximation
Week 5	Non-Negative Matrix Factorization (NMF) and Its Variants
Week 6	Basic Tensor Operations: Folding, Unfolding (Matricization), and Vectorization
Week 7	Tensor Products and Their Relationships, Kruskal Rank
Week 8	Least Squares, Alternating Least Squares (ALS), and the ADMM Method
Week 9	Tucker Decomposition, CP Decomposition, and Higher-Order SVD (HOSVD)
Week 10	Tensor Train (TT) and Tensor Ring (TR) Decompositions
Week 11	High Dimensional Model Representation (HDMR)
Week 12	Enhanced Multivariance Products Representation (EMPR)
Week 13	Holistic Multivariance Decomposition (HMD)
Week 14	Applications of Tensor Decomposition in Science and Engineering

Primary Textbooks & References

Tensor Decompositions for Data Science

Grey Ballard and Tamara G. Kolda — Cambridge University Press (2025)

Tensor Decompositions and Applications

Tamara G. Kolda and Brett W. Bader — SIAM Review, Vol. 51, No. 3, pp. 455–500 (2009)

Tensor Decomposition for Signal Processing and Machine Learning

Nicholas D. Sidiropoulos, Lieven De Lathauwer et al. — IEEE Transactions on Signal Processing, Vol. 65, No. 13 (2017)

A New Feature Extraction Scheme Based on Support Optimization in Enhanced Multivariance Products Representation for Hyperspectral Imagery

Esen Sen and Süha Tuna — Journal of the Franklin Institute, Vol. 362, Issue 2 (2025)

Low Rank Approximation: Algorithms, Implementation, Applications

Ivan Markovsky — Springer (2012)

Matrix and Tensor Decompositions in Signal Processing

Gérard Favier — Wiley (2021)