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Canonical Polyadic decomposition with a Columnwise Orthonormal Factor Matrix

Abstract : Canonical Polyadic Decomposition (CPD) of a higher-order tensor is an important tool in mathematical engineering. In many applications at least one of the matrix factors is constrained to be column-wise orthonormal. We first derive a relaxed condition that guarantees uniqueness of the CPD under this constraint. Second, we give a simple proof of the existence of the optimal low-rank approximation of a tensor in the case that a factor matrix is column-wise orthonormal. Third, we derive numerical algorithms for the computation of the constrained CPD. In particular, orthogonality-constrained versions of the CPD methods based on simultaneous matrix diagonalization and alternating least squares are presented. Numerical experiments are reported.
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Mikael Sorensen, Lieven de Lathauwer, Pierre Comon, Sylvie Icart, Luc Deneire. Canonical Polyadic decomposition with a Columnwise Orthonormal Factor Matrix. SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 2012, 33 (4), pp.1190-1213. ⟨10.1137/110830034⟩. ⟨hal-00781143⟩

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