Introduction to Differentiation in Matrix Calculus

Introduction to tensor, tensor
factorization and its applications
Mu Li
iPAL Group Meeting
Sept. 17, 2010
 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
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:
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 tensor
 Matrix:
 Not true for higher-order tensor
Tensor factorization: Tucker factorization
 Tucker factorization:
 For three-way tensor, Tucker factorization has three types:
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,
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
 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
 There are still many other tensor factorizations, such as blockoriented decomposition, DEDICOM, CANDELINC.
 Tensor factorizations have wide applications in data
reconstruction, cluster analysis, compression etc.
 Kolda, Bader, Tensor decompositions and applications.
 Martin, an overview of multilinear algebra and tensor
 Cichocki, etc., nonnegative matrix and tensor factorizations.

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