Table 2 Equivalent constrained PARAFAC models

PARAFAC‐3

${\mathbf{X}}_{I_1\times I_2{I}_3}={\mathbf{A}}^{\left(1\right)}{({\mathbf{A}}^{\left(2\right)}◇{\mathbf{A}}^{\left(3\right)})}^T$

Tucker‐3

${\mathbf{X}}_{I_1\times I_2{I}_3}={\mathbf{A}}^{\left(1\right)}{\mathbf{G}}_{R_1\times R_2{R}_3}{\left({\mathbf{A}}^{\left(2\right)}\otimes {\mathbf{A}}^{\left(3\right)}\right)}^T$

Tucker‐(2,3)

$\mathcal{X}={\mathcal{I}}_{3,R}{\times }_1{\mathbf{A}}^{\left(1\right)}{\mathit{\Psi }}^{\left(1\right)}{\times }_2{\mathbf{A}}^{\left(2\right)}{\mathit{\Psi }}^{\left(2\right)}{\times }_3{\mathbf{G}}_{I_3\times R_1{R}_2}$

${\mathbf{X}}_{I_1\times I_2{I}_3}={\mathbf{A}}^{\left(1\right)}{\mathit{\Psi }}^{\left(1\right)}{({\mathbf{A}}^{\left(2\right)}{\mathit{\Psi }}^{\left(2\right)}◇{\mathbf{G}}_{I_3\times R_1{R}_2})}^T$

${\mathit{\Psi }}^{\left(1\right)}={\mathbf{I}}_{R_1}\otimes {\mathbf{1}}_{R_2}^T,{\mathit{\Psi }}^{\left(2\right)}={\mathbf{1}}_{R_1}^T\otimes {\mathbf{I}}_{R_2}$

PARALIND/CONFAC‐3

$\mathcal{X}={\mathcal{I}}_{3,R}{\times }_1{\mathbf{A}}^{\left(1\right)}{\mathit{\Phi }}^{\left(1\right)}{\times }_2{\mathbf{A}}^{\left(2\right)}{\mathit{\Phi }}^{\left(2\right)}{\times }_3{\mathbf{A}}^{\left(3\right)}{\mathit{\Phi }}^{\left(3\right)}$

${\mathbf{X}}_{I_1\times I_2{I}_3}={\mathbf{A}}^{\left(1\right)}{\mathit{\Phi }}^{\left(1\right)}{({\mathbf{A}}^{\left(2\right)}{\mathit{\Phi }}^{\left(2\right)}◇{\mathbf{A}}^{\left(3\right)}{\mathit{\Phi }}^{\left(3\right)})}^T$

= A⁽¹⁾Φ⁽¹⁾(Φ⁽²⁾◇Φ⁽³⁾)^T(A⁽²⁾⊗A⁽³⁾)^T

PARATUCK‐2

$\mathcal{X}={\mathcal{I}}_{3,R}{\times }_1{\mathbf{A}}^{\left(1\right)}{\mathit{\Psi }}^{\left(1\right)}{\times }_2{\mathbf{A}}^{\left(2\right)}{\mathit{\Psi }}^{\left(2\right)}{\times }_3\mathbf{F}$

${\mathit{\Psi }}^{\left(1\right)}={\mathbf{I}}_{R_1}\otimes {\mathbf{1}}_{R_2}^T,{\mathit{\Psi }}^{\left(2\right)}={\mathbf{1}}_{R_1}^T\otimes {\mathbf{I}}_{R_2}$

${\mathbf{X}}_{I_1\times I_2{I}_3}={\mathbf{A}}^{\left(1\right)}{\mathit{\Psi }}^{\left(1\right)}{({\mathbf{A}}^{\left(2\right)}{\mathit{\Psi }}^{\left(2\right)}◇\mathbf{F})}^T$

F=(Φ⁽¹⁾◇Φ⁽²⁾)^Tdiag(vec(C^T))

PARATUCK‐(2,4)

$\mathcal{X}={\mathcal{I}}_{4,R}{\times }_1{\mathbf{A}}^{\left(1\right)}{\mathit{\Psi }}^{\left(1\right)}{\times }_2{\mathbf{A}}^{\left(2\right)}{\mathit{\Psi }}^{\left(2\right)}{\times }_3\mathbf{F}{\times }_4\mathbf{D}$

${\mathbf{X}}_{I_1\times I_2{I}_3{I}_4}={\mathbf{A}}^{\left(1\right)}{\mathit{\Psi }}^{\left(1\right)}{({\mathbf{A}}^{\left(2\right)}{\mathit{\Psi }}^{\left(2\right)}◇\mathbf{F}◇{\mathbf{C}}_{I_4\times R_1{R}_2})}^T$

${\mathit{\Psi }}^{\left(1\right)}={\mathbf{I}}_{R_1}\otimes {\mathbf{1}}_{R_2}^T,{\mathit{\Psi }}^{\left(2\right)}={\mathbf{1}}_{R_1}^T\otimes {\mathbf{I}}_{R_2}$

${\mathbf{X}}_{I_1{I}_2\times I_3{I}_4}=\left({\mathbf{A}}^{\left(1\right)}\otimes {\mathbf{A}}^{\left(2\right)}\right)$

$\mathbf{F}={({\mathit{\Phi }}^{\left(1\right)}◇{\mathit{\Phi }}^{\left(2\right)})}^T,\mathbf{D}={\mathbf{C}}_{I_4\times R_1{R}_2}$

${\left({({\mathit{\Phi }}^{\left(1\right)}◇{\mathit{\Phi }}^{\left(2\right)})}^T◇{\mathbf{C}}_{I_4\times R_1{R}_2}\right)}^T$