Overview of constrained PARAFAC models

Favier, Gérard; de Almeida, André LF

doi:10.1186/1687-6180-2014-142

EURASIP Journal on Advances in Signal Processing

Table 1 Main tensor models

From: Overview of constrained PARAFAC models

Models	Scalar writings	Mode‐ n product‐based writings
PARAFAC‐3	$x_{i_{1}, i_{2}, i_{3}} = \sum_{r}^{R} a_{i_{1}, r}^{(1)} a_{i_{2}, r}^{(2)} a_{i_{3}, r}^{(3)}$	$X = I_{3, R} \times_{1} A^{(1)} \times_{2} A^{(2)} \times_{3} A^{(3)}$
Tucker‐3	$x_{i_{1}, i_{2}, i_{3}} = \sum_{r_{1} = 1}^{R_{1}} \sum_{r_{2} = 1}^{R_{2}} \sum_{r_{3} = 1}^{R_{3}} g_{r_{1}, r_{2}, r_{3}} a_{i_{1}, r_{1}}^{(1)} a_{i_{2}, r_{2}}^{(2)} a_{i_{3}, r_{3}}^{(3)}$	$X = G \times_{1} A^{(1)} \times_{2} A^{(2)} \times_{3} A^{(3)}$
Tucker‐(2,3)	$x_{i_{1}, i_{2}, i_{3}} = \sum_{r_{1} = 1}^{R_{1}} \sum_{r_{2} = 1}^{R_{2}} g_{r_{1}, r_{2}, i_{3}} a_{i_{1}, r_{1}}^{(1)} a_{i_{2}, r_{2}}^{(2)}$	$X = G \times_{1} A^{(1)} \times_{2} A^{(2)}$
PARALIND/CONFAC‐3	$x_{i_{1}, i_{2}, i_{3}} = \sum_{r_{1} = 1}^{R_{1}} \sum_{r_{2} = 1}^{R_{2}} \sum_{r_{3} = 1}^{R_{3}} g_{r_{1}, r_{2}, r_{3}} a_{i_{1}, r_{1}}^{(1)} a_{i_{2}, r_{2}}^{(2)} a_{i_{3}, r_{3}}^{(3)}$	$X = G \times_{1} A^{(1)} \times_{2} A^{(2)} \times_{3} A^{(3)}$
	$g_{r_{1}, r_{2}, r_{3}} = \sum_{r}^{R} φ_{r_{1}, r}^{(1)} φ_{r_{2}, r}^{(2)} φ_{r_{3}, r}^{(3)}$	$G = I_{3, R} \times_{1} Φ^{(1)} \times_{2} Φ^{(2)} \times_{3} Φ^{(3)}$
PARATUCK‐2	$x_{i_{1}, i_{2}, i_{3}} = \sum_{r_{1} = 1}^{R_{1}} \sum_{r_{2} = 1}^{R_{2}} g_{r_{1}, r_{2}, i_{3}} a_{i_{1}, r_{1}}^{(1)} a_{i_{2}, r_{2}}^{(2)}$	$X = G \times_{1} A^{(1)} \times_{2} A^{(2)}$
	$g_{r_{1}, r_{2}, i_{3}} = c_{r_{1}, r_{2}} φ_{r_{1}, i_{3}}^{(1)} φ_{r_{2}, i_{3}}^{(2)}$	$G = I_{3, R} \times_{1} Ψ^{(1)} \times_{2} Ψ^{(2)} \times_{3} F$ , R=R₁R₂
		$Ψ^{(1)} = I_{R_{1}} \otimes 1_{R_{2}}^{T}, Ψ^{(2)} = 1_{R_{1}}^{T} \otimes I_{R_{2}}$
		F=(Φ⁽¹⁾◇Φ⁽²⁾)^Tdiag(vec(C^T))
PARATUCK‐(2,4)	$x_{i_{1}, i_{2}, i_{3}, i_{4}} = \sum_{r_{1} = 1}^{R_{1}} \sum_{r_{2} = 1}^{R_{2}} g_{r_{1}, r_{2}, i_{3}, i_{4}} a_{i_{1}, r_{1}}^{(1)} a_{i_{2}, r_{2}}^{(2)}$	$X = G \times_{1} A^{(1)} \times_{2} A^{(2)}$
	$g_{r_{1}, r_{2}, i_{3}, i_{4}} = c_{r_{1}, r_{2}, i_{4}} φ_{r_{1}, i_{3}}^{(1)} φ_{r_{2}, i_{3}}^{(2)}$	$G = I_{4, R} \times_{1} Ψ^{(1)} \times_{2} Ψ^{(2)} \times_{3} F \times_{4} D$ , R=R₁R₂
		$Ψ^{(1)} = I_{R_{1}} \otimes 1_{R_{2}}^{T}, Ψ^{(2)} = 1_{R_{1}}^{T} \otimes I_{R_{2}}$
		$F = {(Φ^{(1)} ◇ Φ^{(2)})}^{T}, D = C_{I_{4} \times R_{1} R_{2}}$

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