Decomposition for a Quaternion Tensor Triplet with Applications

He, Zhuo-Heng; Navasca, Carmeliza; Wang, Xiangxiang

doi:10.1007/s00006-021-01195-8

Cited by 12 publications

(1 citation statement)

References 47 publications

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“…The lack of such framework can be explained by the noncommutativity of the quaternion product, which prevents a direct extension of classical real and complex tensor definitions. A few attempts at defining tensor-like tools for quaternion multiway arrays have been made recently [9][10][11][12][13]. However, a proper definition of quaternion tensors based on multilinear arguments remains Work supported by the CNRS/INS2I through the TenQ project (2022).…”

Section: Introductionmentioning

confidence: 99%

Low-Rank Tensor Decompositions for Quaternion Multiway Arrays

Imhogiemhe,

Flamant,

Luciani

et al. 2023

ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Quaternion multiway arrays appear naturally as compact representations of 3D or 4D multidimensional signals. However, the non-commutativity of quaternion multiplication prevents a straightforward extension of standard tensor algebra to analyze and process quaternion multiway arrays. After reviewing the theoretical difficulties related to quaternion tensor algebra, we propose the first construction of quaternion tensors as representation of dedicated quaternion multilinear forms. This theoretical construction ensures that usual tensor algebraic properties, such as mode products properties are preserved. This novel framework enables us to generalize Tucker and canonical polyadic tensor decompositions to the quaternion case. For the latter, we carefully design a full quaternion ALS-type algorithm. Its relevance is validated numerically.

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