1. CIS 5590: Large-Scale Matrix Decomposition Tensors and Applications
Instructor: Kai Zhang
Temple University, Spring 2018

2. What is a Tensor? A tensor is a multi-way extension of a matrix
A multi-dimensional array
A multi-linear map
Examples
Scalar, Vector, Matrices
color images, videos
data matrices, recorded at different times t
ECG: electrode x ECG beat x time
action recognition

4. Epilespy Sedure Detection

5. Fibers and Slices
Third
First
Second

6. Tensor Matrication Definition: unfolds an N-way tensor into a matrix
Mode-n matricization arranges the mode-n fibers as columns of a matrix Denoted X(n)
As many rows as is the dimensionality of the nth mode
As many columns as is the product of the dimensions of other modes

7. Tensor Multiplication
Definition

8. Pictorial view of Tensor Multiplication

9. Mode-n Product Example

10. Kronecker Product

11. Khatri–Rao Matrix Product

12. Khatri–Rao Matrix Product

13. Tensor Decomposition
CP decomposition

14. Tensor Decomposition
Using matricization, we can re-write the CP decomposition
ALS Algorithm

15. Tucker Decomposition

17. ALS Solution for Turker

18. Tucker Decomposition