## MATLAB Tensor Toolbox Version 2.0

Tensors (also known as mutidimensional arrays or N-way arrays) are used in a variety of applications ranging from chemometrics to psychometrics. The MATLAB tensor toolbox provides the following classes for manipulating dense, sparse, and structured tensors using MATLAB's object-oriented features:

• tensor - An extension of MATLAB's native multidimensional array capabilities.
• sptensor - A sparse multidimensional array.
• tenmat - Store a tensor as a matrix, with extra information so that it can be converted back into a tensor.
• sptenmat - Store a sptensor as sparse matrix in coordinate format, with extra information so that it can be converted back into a sptensor.
• ttensor - Store a tensor decomposed as a Tucker operator (see below).
• ktensor - Store a tensor decomposed as a Kruskal operator (see below).

### Glossary

A sparse tensor is a tensor where only a small fraction of the elements are nonzero. In this case, it is more efficient to store just the nonzeros and their indices.

A tensor that is decomposed as a Tucker Operator comprises a core tensor multiplied in each mode by a matrix. For a three-way array, this means the tensor X can be written as:

xijk = Σr Σs Σt grst air bjs ckt for all i,j,k

Thus, the tensor X may be stored in terms of its components, the core tensor G and the factor matrices A,B,C.

A tensor that is decomposed as a Kruskal Operator comprises a component matrix for each mode and an optional scaling vector. For a three-way array, this means the tensor X can be written as:

xijk = Σr λr air bjr ckr for all i,j,k.

Thus, the tensor X may be stored in terms of its components, the vector λ and the factor matrices A,B,C.

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### Related Papers

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Contacts
Tamara Kolda
(tgkolda@sandia.gov)
(925)294-4769