Introduction to Differentiation in Matrix Calculus

Mu LiiPAL Group MeetingSept. 17, 2010Introduction to tensor, tensor factorization and its applicationsOutline Basic concepts about tensor 1. Whats tensor? Why tensor and tensor factorization? 2. Tensor multiplication 3. Tensor rankTensor factorization 1. CANDECOMP/PARAFAC factorization 2. Tucker factorizationApplications of tensor factorizationConclusionWhats 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:

A tensor is a multidimensional array

Fiber and slice

Tensor unfoldings: Matricization and vectorization

Matricization: convert a tensor to a matrix

Vectorization: convert a tensor to a vector

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

Definition:

Tensor multiplication: the n-mode product: multiplied by a vector

Definition:

Note: multiplying by a vector reduces the dimension by one.

Rank-one Tensor and Tensor rank

Rank-one tensor: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.

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.

Differences between matrix SVD and tensor CP

Lower-rank approximation is different between matrix and higher-order tensorMatrix:

Not true for higher-order tensor

Tensor factorization: Tucker factorization

Tucker factorization:

For three-way tensor, Tucker factorization has three types:Tucker3:

Tucker2:

Tucker1:

Three types of Tucker factorization

Tucker factorization

Uniqueness: Unlike CP, Tucker factorization is not unique.

How to compute:

Higher-order SVD(HOSVD), for each n,

Rn:

Applications of Tensor factorization

A simple application of CP:

Apply CP to reconstruct a MATLAB logo from noisy data

Apply Tucker3 to do data reconstruction from noise

Apply Tucker3 to do cluster analysis

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.

References

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