. Author manuscript; available in PMC: 2023 Nov 1.

Published in final edited form as: IEEE Trans Knowl Data Eng. 2020 Dec 21;34(10):4854–4873. doi: 10.1109/tkde.2020.3045924

TABLE 3:

A summary of relation-learning based HNE algorithms. (Additional notations: $f$ : a non-linear activation function; ${[x]}_{+} : \max {x, 0}; ∥ \cdot ∥_{F}$ : the Frobenius norm of a matrix; $𝓕, 𝓕^{- 1}$ : the fast Fourier transform and its inverse; $\bar{x}$ : the complex conjugate of $x$ ; $l^{- 1}$ : the inverse of relation $l$ ; $\times_{i}$ : the tensor product along the $i$ -th mode; $Θ$ : the set of learned parameters; $E_{u}, E_{l}$ : 2D reshaping matrices of $e_{u}$ and $e_{l}$ [82]; vec: the vectorization operator; *: the convolution operator; $M (e_{u}, e_{l})$ : a matrix aligning the output vectors from the convolution with all kernels [84].)

Algorithm	$w_{u v}^{(l)}$	$d (e_{u}, e_{v})$	$𝓙_{R 0}$	Applications
TransE [4]	$1_{(u, v) \in E_{l}}$	$‖ e_{u} + e_{l} - e_{v} ‖$	$\sum_{v} (‖ e_{v} ‖ - 1)$	KB completion, relation extraction from text
TransH [118]		$\begin{matrix} {‖ e_{u, l} + e_{l} - e_{v, l} ‖}^{2}, \\ e_{v, l} = e_{v} - w_{l}^{T} e_{v} w_{l} \end{matrix}$	$\begin{matrix} \sum_{l} (‖ w_{l} ‖ - 1) + \sum_{v} {[∥ e_{v} ∥ - 1]}_{+} \\ + \sum_{l} {[\frac{{(w_{l}^{T} e_{l})}^{2}}{{‖ e_{l} ‖}^{2}} - ϵ^{2}]}_{+} \end{matrix}$
TransR [80]		$\begin{matrix} {‖ e_{u, l} + e_{l} - e_{v, l} ‖}^{2}, \\ e_{v, l} = A_{l} e_{v} \end{matrix}$	$\begin{matrix} \sum_{v} {[‖ e_{v} ‖ - 1]}_{+} + \sum_{l} {[‖ e_{l} ‖ - 1]}_{+} \\ + \sum_{v} {[‖ A_{l} e_{v} ‖ - 1]}_{+} \end{matrix}$
RHINE [76]	edge weight of $(u, v)$ with type $l$	${‖ e_{u} - e_{v} ‖}^{2}$ if $l$ models affiliation, $‖ e_{u} + e_{l} - e_{v} ‖$ if $l$ models interaction	N/A	link prediction, node classification
RotatE [114]	$1_{(u, v) \in E_{l}}$	${‖ e_{u} ⊙ e_{l} - e_{v} ‖}^{2}, e_{u}, e_{v}, e_{l} \in ℂ^{n}$	$\sum_{l} (‖ e_{l} ‖ - 1)$	KB completion, relation pattern inference
RESCAL [119]		${‖ \sqrt{A_{l}} (e_{u} - e_{v}) ‖}^{2}$	$\sum_{v} {[‖ e_{v} ‖ - 1]}_{+} + \sum_{l} {[{‖ A_{l} ‖}_{F} - 1]}_{+}$	entity resolution, link prediction
DistMult [81]		${‖ \sqrt{A_{l}} (e_{u} - e_{v}) ‖}^{2}, A_{l} = d i a g (e_{l})$	$\sum_{v} (‖ e_{v} ‖ - 1) + \sum_{l} {[‖ e_{l} ‖ - 1]}_{+}$	KB completion, triplet classification
HolE [120]		${‖ e_{r} - 𝓕^{- 1} (\bar{𝓕 (e_{u})} ⊙ 𝓕 (e_{v})) ‖}^{2}$	$\sum_{v} {[‖ e_{v} ‖ - 1]}_{+} + \sum_{l} {[‖ e_{l} ‖ - 1]}_{+}$
ComplEx [121]		${‖ A_{l} e_{u} - e_{v} ‖}^{2}, A_{l} = d i a g (e_{l}), e_{u}, e_{v}, e_{l} \in ℂ^{n}$	$\sum_{v} {‖ e_{v} ‖}^{2} + \sum_{l} {‖ e_{l} ‖}^{2}$
SimplE [122]		$\begin{matrix} \frac{1}{2} ({‖ \sqrt{A_{l}} (e_{u} - e_{v}) ‖}^{2} + {‖ \sqrt{A_{l - 1}} (e_{v} - e_{u}) ‖}^{2}), \\ A_{l} = d i a g (e_{l}), A_{l - 1} = d i a g (e_{l - 1}) \end{matrix}$	$\sum_{v} {‖ e_{v} ‖}^{2} + \sum_{l} {‖ e_{l} ‖}^{2}$
TuckER [123]		$C - 𝓦 \times_{1} e_{u} \times_{2} e_{l} \times_{3} e_{v}$	N/A
NTN [20]		$C - e_{l}^{T} \tanh (e_{u}^{T} 𝓜_{l} e_{v} + M_{l, 1} e_{u} + M_{l, 2} e_{v} + b_{l})$	$∥ Θ ∥_{2}^{2}$
ConvE [82]		$C - σ (vec (σ ([E_{u}; E_{l}] * ω)) W) e_{v}$	N/A
NKGE [83]		same as TransE or ConvE, where $e_{u} = σ (g_{u}) ⊙ e_{u}^{s} + (1 - σ (g_{u})) ⊙ e_{u}^{n}$	$∥ Θ ∥_{2}^{2}$
SACN [84]		$C - f (vec (M (e_{u}, e_{l})) W) e_{v}$	N/A