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:

x_ijk = Σ_r Σ_s Σ_t g_rst a_ir b_js c_kt 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:

x_ijk = Σ_r λ_r a_ir b_jr c_kr 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.

Download

To download the software, proceed first to the Tensor Toolbox License and Registration page.

Mailing List

Please join our Tensor Toolbox Mailing List to keep updated on the latest releases and uses for the MATLAB Tensor Toolbox.

How to Cite

Please cite the following two references for the MATLAB Tensor Toolbox Version 2.1.

• Brett W. Bader and Tamara G. Kolda, Efficient MATLAB computations with
  sparse and factored tensors, Technical Report SAND2006-7592, Sandia National Laboratories, Albuquerque, NM and Livermore, CA, 2006.
• Brett W. Bader and Tamara G. Kolda, MATLAB Tensor Classes for Fast Algorithm Prototyping, ACM Transactions on Mathematical Software, 32(4), December 2006..
• Brett W. Bader and Tamara G. Kolda, MATLAB Tensor Toolbox Version 2.1, http://csmr.ca.sandia.gov/~tgkolda/TensorToolbox/, December 1, 2006.

Questions or Comments

• Send a question or comment on the Tensor Toolbox via email
• Report a bug in Bugzilla (requires registration)

Related Papers

Do you have a paper that uses the MATLAB Tensor Toolbox? If so, let us know and we'll post it here. Thanks!

• E. Acar, S. A. Çamtepe, M. Krishnamoorthy and B. Yener, Modeling and
  Multiway Analysis of Chatroom Tensors, Proc. of IEEE International
  Conference on Intelligence and Security Informatics, LNCS, Vol. 3495,
  Kantor P. and Muresan G.; Roberts, F.; et al. (Eds.), pp 256-268, 2005.
• Brett W. Bader, Richard Harshman, and Tamara G. Kolda. Temporal analysis of social networks using three-way DEDICOM. Technical Report SAND2006-2161, Sandia National Laboratories, Albuquerque, NM and Livermore, CA, April 2006.
• Daniel M. Dunlavy, Tamara G. Kolda, and W. Philip Kegelmeyer. Multilinear algebra for analyzing data with multiple linkages. Technical Report SAND2006-2079, Sandia National Laboratories, Albuquerque, NM and Livermore, CA, April 2006.
• D. FitzGerald, M. Cranitch, and E. Coyle, Shifted Non-negative Matrix Factorisation for Sound Source Separation, Proc. IEEE Conf. on Statistics in Signal Processing, Bordeaux, France, July 2005.
• D. FitzGerald, M. Cranitch, and E. Coyle., Non-negative Tensor Factorisation for Sound Source Separation, Proc. Irish Signals and Systems Conf., Dublin, September 2005.
• D. FitzGerald, M. Cranitch, and E. Coyle, Sound Source Separation using shifted Non-negative Tensor Factorisation, Proc. ICASSP06, Toulouse, France, 2006.
• D. FitzGerald, M. Cranitch, and E. Coyle, Shifted 2D Non-negative Tensor Factorisation, Proc. Irish Signals and Systems Conference, Dublin, June 2006.
• Tamara Kolda and Brett Bader. The TOPHITS model for higher-order web link analysis. In Workshop on Link Analysis, Counterterrorism and Security, 2006.
• Tamara G. Kolda, Brett W. Bader, and Joseph P. Kenny. Higher-order web link analysis using multilinear algebra. In ICDM 2005: Proceedings of the 5th IEEE International Conference on Data Mining, pages 242–249, November 2005.
• Jimeng Sun, Dacheng Tao, and Christos Faloutsos. Beyond streams and graphs: dynamic tensor analysis. In KDD '06: Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 374–383, 2006.

Links to Previous Versions

• Version 2.0
• Version 1.0