TA stands for tensor approximation and it is a mathematical framework, which is used in various areas for compact data representation. Here a couple of helpful links on the basic definitions, models and algorithms are given.

Books, Reviews and Surveys

• Kolda & Bader, 2009: Tensor Decompositions and Applications
• De Lathauwer, 2009: A Survey of Tensor Methods
• Kroonenberg, 2008: Applied Multiway Data Analysis
• Smilde et al., 2004: Multi-way Analysis: Applications in the Chemical Sciences
• Moravitz, 2004: Tensor decompositions workshop discussion notes

Theory and Background

Notation

• Kolda & Bader, 2009: Tensor Decompositions and Applications
• Smilde et al., 2004: Multi-way Analysis: Applications in the Chemical Sciences
• Kiers, 2000: Towards a standardized notation and terminol- ogy in multiway analysis
• De Lathauwer et al, 2000b: On the best rank-1 and rank-(R1,R2,...,RN) approximation of higher-order tensors
• De Lathauwer et al, 2000a: A multilinear singular value decomposition

Models

• Tucker model: Tucker, 1963: Implications of Factor Analysis of Three-way Matrices for Measurement of Change, Tucker, 1964: The Extension of Factor Analysis to Three-dimensional Matrices, and Tucker, 1966: Some Mathematical Notes on Three-mode Factor Analysis
• CP model: Caroll & Chang, 1970: Analysis of Individual Differences in Multidimensional Scaling via an N-way Generalization of Eckart--Young Decompositions and Harshman, 1970: Foundations of the PARAFAC Procedure: Models and Conditions for an Explanatory Multi--modal Factor Analysis
• Block-based TA: De Lathauwer, 2008: Decompositions of a Higher-Order Tensor in Block Terms---Part II: Definitions and Uniqueness
• Others see Kolda & Bader, 2009: Tensor Decompositions and Applications

Algorithms

Alternating least-squares (ALS) algorithms:

• Tucker1 (three-mode SVD): Tucker, 1966: Some Mathematical Notes on Three-mode Factor Analysis
• Three-mode PCA ALS: Kroonenberg & De Leeuw, 1980: Principal Component Analysis of Three-mode Data by Means of Alternating Least Squares Algorithms and Kroonenberg, 1983: Three-mode principal component analysis: Theory and applications
• TUCKALS: Ten Berge et al., 1987: Some Additional Results on Principal Components Analysis of Three-mode Data by Means of Alternating Least Squares Algorithms
• Higher-order singular value decomposition (HOSVD): De Lathauwer et al, 2000a: A multilinear singular value decomposition
• Higher-order orthogonal iteration (HOOI) and higher-order power method (HOPM): De Lathauwer et al, 2000b: On the best rank-1 and rank-(R1,R2,...,RN) approximation of higher-order tensors
• Block-based TA ALS: De Lathauwer & Nion, 2008: Decompositions of a Higher-Order Tensor in Block Terms---Part III: Alternating Least Squares Algorithms
• For improvements on the above algorithms see literature in Kolda & Bader, 2009: Tensor Decompositions and Applications

Libraries

• Tensor classes in vmmlib
• Tensor Toolbox for MATLAB
• Tensorlab
• Others see Kolda & Bader, 2009: Tensor Decompositions and Applications