关系张量分解：多关系链接预测的新方法
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本文主要探讨了在多关系知识图谱（KG）中进行链接预测的一种新型方法——关系张量分解（Relational Tucker，简称RT）。关系张量分解是一种矩阵和张量的乘积表示形式，它将实体嵌入和关系嵌入的概念进行分离，打破了传统知识图谱模型中实体和关系嵌入大小固定的限制。RT模型允许不同关系之间参数共享，这意味着它可以更灵活地处理多对多的关系结构，而无需预设的稀疏性模式。
与现有的知识图谱嵌入模型（如TransE, DistMult, ComplEx等）相比，RT模型并不依赖于特定的稀疏性策略，而是提供了一个探索自动学习稀疏性模式的可能性。作者试图回答两个关键问题：一是能否通过RT分解自动学习出最佳的稀疏性模式，二是对于相同的参数数量，密集型模型（全连接模型）是否能够优于稀疏型模型。实验结果显示，答案取决于具体的数据集，某些情况下，自动学习到的稀疏性可以提高模型性能，而密集模型在某些条件下确实可以取得更好的预测效果。
RT分解的提出是针对多关系知识图谱中的复杂链接预测任务，它扩展了传统的知识表示框架，使得模型设计更具灵活性。它不仅提升了模型的适应性和泛化能力，而且为模型选择和优化提供了新的途径。通过对比实验，研究者们展示了RT分解作为一种强大的工具，可以帮助我们更好地理解在知识图谱建模中稀疏性与模型性能之间的关系，以及如何根据数据特性来调整模型架构。
这篇文章对关系张量分解在多关系知识图谱中的应用进行了深入研究，为未来知识图谱的链接预测和模型优化提供了新的思考角度。对于那些致力于提升知识图谱理解和预测准确性的研究人员来说，这篇文章的价值在于提供了实用的理论支持和实验依据，有助于推动该领域的进一步发展。
A Relational Tucker Decomposition for Multi-Relational Link Prediction
where
Re(·)
extracts the real part of a complex number.
Likewise, HolE (Nickel et al., 2016b) is equivalent to Com-
plEx (Hayashi & Shimbo, 2017). HolE uses the scoring
function
s(i,k, j) = r
T
k
(e
i
? e
j
)
, where
?
refers to the circu-
lar correlation between
e
i
and
e
j
∈ R
d
e
(i.e.,
(e
i
? e
j
)
k
=
∑
d
r
t=1
e
it
e
j((k+t−2 mod d
r
)+1)
). The idea of using circular cor-
relation relates to associative memory (Nickel et al., 2016b).
HolE uses as
M
HolE
k
the circumvent matrix resulting from
r
k
. In the remainder of this paper, we use the formulation
using M
ComplEx
k
given above.
Analogy (Liu et al., 2017b).
Analogy uses block-
diagonal mixing matrices
M
Analogy
k
, where each block
is either (1) a real scalar
x
or (2) a
2 × 2
matrix of
form
x −y
y x
, where
x,y ∈ R
refer to entries of
r
k
and
each entry of
r
k
appears in exactly one block. We have
d
r
= d
e
. Analogy aims to capture commutative relational
structure: the constraint ensures that
M
Analogy
k
1
M
Analogy
k
2
=
M
Analogy
k
2
M
Analogy
k
1
for all
k
1
,k
2
∈ R
. Both DistMult (only
first case allowed) and ComplEx (only second case allowed)
are special cases of Analogy.
3. The Relational Tucker3 Decomposition
In this section, we introduce the Relational Tucker3 (RT) de-
composition, which decomposes the KG into entity embed-
dings, relation embeddings, and a core tensor. We show that
each of the existing bilinear models can be viewed (1) as an
unconstrained RT decomposition with a fixed (sparse) core
tensor or (2) as a constrained RT decomposition with fixed
relation embeddings. In contrast to bilinear models, the RT
decomposition allows parameters sharing across different
relations, and it decouples the entity and relation embedding
sizes. Our experimental study (4) suggests that both proper-
ties can be beneficial. We also introduce a sparse variant of
RT called SRT to approach the question of whether we can
learn sparsity patterns of the core tensor from the data.
In what follows, we make use of the tensor representation
of KGs. In particular, we represent knowledge graph
K
via a binary tensor
X ∈ {0, 1}
N×N×K
, where
x
i jk
= 1
if
and only if
(i,k, j) ∈ K
. Note that if
x
i jk
= 0
, we assume
that the truth value of triple
(i,k, j)
is missing instead of
false. The scoring tensor of a particular embedding model
m
of
K
is the tensor
S ∈ R
N×N×K
of all predicted scores,
i.e., with
s
i jk
= s
m
(i,k, j)
. We use
A
k
to refer to the
k
-th
frontal slice of a 3-way tensor
A
. Note that
X
k
contains the
data for relation
k
, and that scoring matrix
S
k
contains the
respective predicted scores. Generally, embedding models
aim to construct scoring tensors
S
that suitably approximate
X (Nickel et al., 2016a).
3.1. Definition
We start with the classicial Tucker3 decomposition (Tucker,
1966), focusing on 3-way tensors throughout. The Tucker3
decomposition decomposes a given data tensor into three
factor matrices (one per mode) and a core tensor, which
stores the weights of the three-way interactions. The de-
composition can be can be viewed as a form of higher-order
PCA (Kolda & Bader, 2009). In more detail, given a tensor
D ∈ R
I×J×K
and sufficiently large parameters
d
a
,d
b
,d
c
∈ N
,
the Tucker3 decomposition factorizes
D
into factor matri-
ces
A ∈ R
I×d
a
,
B ∈ R
J×d
b
,
C ∈ R
K×d
c
, and core tensor
H ∈ R
d
a
×d
b
×d
c
such that
d
i jk
= a
T
i
[H ×
3
c
k
]b
j
,
where
H×
3
c
k
∈ R
d
a
×d
b
refers to the mode-3 tensor product
defined as
H ×
3
c
k
=
d
c
∑
l=1
c
kl
H
l
,
i.e., a linear combination of the frontal slices of
H
. If
d
a
,d
b
,d
c
are smaller than
I,J,K
, core tensor
H
can be inter-
preted as a compressed version of
D
. It is well-known that
the CP decomposition (Kolda & Bader, 2009) corresponds
to the special case where
d
a
= d
b
= d
c
and
H
is fixed to
the
d
a
× d
a
× d
a
tensor with
h
i jk
= 1
iff
i = j = k
, else 0.
The RT decomposition, which we introduce next, allows us
to view existing bilinear models as decompositions with a
fixed core tensor as well.
In particular, in KG embedding models, we associate each
entity with a single embedding, which we use to represent
the entity in both subject and object position. The relational
Tucker3 (RT) decomposition applies this approach to the
Tucker3 decomposition by enforcing
A = B
. In particular,
given embedding sizes
d
e
and
d
r
, the RT decomposition is
parameterized by an entity embedding matrix
E ∈ R
N×d
e
,
a relation embedding matrix
R ∈ R
K×d
r
, and a core tensor
G ∈ R
d
e
×d
e
×d
r
. As in the standard Tucker3 decomposition,
RT composes mixing matrices from the frontal slices of the
core tensor, i.e.,
M
RT
k
=
d
r
∑
l=1
r
kl
G
l
.
The scoring tensor has entries
s
i jk
= e
T
i
(G ×
3
r
k
)e
j
= e
T
i
M
RT
k
e
j
. (2)
Note that the mixing matrices for different relations share
parameters through the frontal slices of the core tensor.
The RT decomposition can represent any given tensor, i.e.,
the restriction on a single embedding per entity does not
limit expressiveness. To see this, suppose that we are given
a Tucker3 decomposition
A,B,C,H
of some tensor. Now
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