1. Introduction to tensor, tensor factorization and its applications
Mu Li
iPAL Group Meeting
Sept. 17, 2010

2. Outline Basic concepts about tensor
1. What’s tensor? Why tensor and tensor factorization?
2. Tensor multiplication
3. Tensor rank
Tensor factorization
1. CANDECOMP/PARAFAC factorization
2. Tucker factorization
Applications of tensor factorization
Conclusion

3. What’s tensor? Why tensor and tensor factorization?
Definition: a tensor is a multidimensional array which is an extension of matrix.
Tensor can happen in daily life.
In order to facilitate information mining from tensor and tensor processing, storage, tensor factorization is often needed.
Three-way tensor:

4. A tensor is a multidimensional array

5. Fiber and slice

6. Tensor unfoldings: Matricization and vectorization
Matricization: convert a tensor to a matrix
Vectorization: convert a tensor to a vector

7. Tensor multiplication: the n-mode product: multiplied by a matrix
Definition:

8. Tensor multiplication: the n-mode product: multiplied by a vector
Definition:
Note: multiplying by a vector reduces the dimension by one.

9. Rank-one Tensor and Tensor rank
Example:
Tensor rank: smallest number of rank-one tensors that can generate it by summing up.
Differences with matrix rank:
1. tensor rank can be different over R and C.
2. Deciding tensor rank is an NP problem that no straightforward algorithm can solve it.

10. Tensor factorization: CANDECOMP/PARAFAC factorization(CP)
Tensor factorization: an extension of SVD and PCA of matrix.
CP factorization:
Uniqueness: CP of tensor(higher-order) is unique under some general conditions.
How to compute:
Alternative Least Squares(ALS), fixing all but one factor matrix to which LS is applied.

11. Differences between matrix SVD and tensor CP
Lower-rank approximation is different between matrix and higher-order tensor
Matrix:
Not true for higher-order tensor

12. Tensor factorization: Tucker factorization
For three-way tensor, Tucker factorization has three types:
Tucker3:
Tucker2:
Tucker1:

13. Three types of Tucker factorization

14. Uniqueness: Unlike CP, Tucker factorization is not unique.
How to compute:
Higher-order SVD(HOSVD), for each n,
Rn:

15. Applications of Tensor factorization
A simple application of CP:

16. Apply CP to reconstruct a MATLAB logo from noisy data

17. Apply Tucker3 to do data reconstruction from noise

18. Apply Tucker3 to do cluster analysis

19. Conclusion
Tensor is a multidimensional array which is an extension of matrix that arises frequently in our daily life such as video, microarray data, EEG data, etc.
Tensor factorization can be considered higher-order generalization of matrix SVD or PCA, but they also have much differences, such as NP essential of deciding higher-order tensor rank, non-optimal property of higher-order tensor factorization.
There are still many other tensor factorizations, such as block- oriented decomposition, DEDICOM, CANDELINC.
Tensor factorizations have wide applications in data reconstruction, cluster analysis, compression etc.

20. Kolda, Bader, Tensor decompositions and applications.
References
Kolda, Bader, Tensor decompositions and applications.
Martin, an overview of multilinear algebra and tensor decompositions.
Cichocki, etc., nonnegative matrix and tensor factorizations.