The decompositions of higher-order tensors can be viewed as generalizations of matrix decompositions to orders higher than two. Figure 1 depicts the parallel factor (PARAFAC) decomposition, also known as the canonical polyadic decomposition (CPD), perhaps the most popular tensor decomposition. It has been extensively studied and considered in numerous application domains, ranging from psychometrics and chemometrics to signal processing and machine learning.

Figure 1: The PARAFAC decomposition of a third-order tensor, represented as a sum of rank-one components.

The attractive feature of the PARAFAC decomposition is its intrinsic uniqueness. In contrast to matrix (bilinear) decompositions, where there is the well-known problem of rotational freedom, the PARARAC decomposition of higher-order tensors is essentially unique, up to scaling and permutation indeterminacies. Tensor decompositions fall within an interdisciplinary research field. Although important progress has been made, research has several intriguing and open-ranging fields ranging from fundamental studies such as uniqueness, degeneracies, and rank to more practical aspects, where tensor decompositions are used to model complex physical phenomena.

Recent attention has also been devoted to distributed algorithms for decomposing large-scale (big data) tensors. In this context, we highlight the role of tensor networks, a network model formed by multiple tensors following specific contracted rules.

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