RESCAL

A Three-Way Model for Collective Learning on Multi-Relational Data

2011

we propose the relational learning approach RESCAL which is based on a tensor factorization that is related to DEDICOM but does not exhibit the same constraints.

1 Introduction

说明最近tensor decomposition方法逐渐被应用在relational learning，原因：

• 从建模的角度讲，tensor decomposition更直接，各种relation可以直接表示为high-order tensor。同时，没有先验知识需要
• 从learning的角度讲，关系型的数据通常是高维并且稀疏的，适用于tensor decomposition。

关系型数据的重要特征是相关性能够通过各种相连的node产生，但是目前的模型都无法很好的满足要求。

we propose the relational learning approach RESCAL which is based on a tensor factorization that is related to DEDICOM but does not exhibit the same constraints.

2 Modelling and Notation

看一下对于relational data如何表示：

整体的数据被表示为张量\(\cal{X}\)，\(\cal{X}_{ijk}=1\)，表示fact存在。

4 Methods and Theoretical Aspects

文中定义了collective learning，大概含义是通过利用相关node的信息perform task。

We will refer to the mechanism of exploiting the information provided by related entities regardless of the particular learning task at hand as collective learning.

4.1 A Model for Multi-Relational Data

核心在于： \[ {\cal{X}_k} \approx AR_kA^T,\ k = 1,2,\dots,m,\\ A \in R^{n\times r}, R_k\in R^{r\times r} \] 对于该式子的理解是，\(R_k\)表示关系\(r\)的转换，\(R_kA^T\)将\(A\)转换到了\(R_k\)表示的向量空间当中，通过与\(A\)乘积，最终得到的对于\((h,r,t)\)，通过点积表示fact。

通过最小化得到最终的embedding： \[ min\ f(A,R_k)+g(A,R_k) \\ f(A,R_k)=\frac{1}{2}(\sum_k {\lVert {\cal{X}}-AR_kA^T \rVert}_F^2) \\ g(A,R_k)=\frac{1}{2}\lambda({\lVert A \rVert}^2_F + \sum_k{\lVert R_k \rVert}^2_F) \]

5 Evaluation

进行了四方面的比较，

• Collective Classification
• Collective Entity Resolution
• Kinships, Nations and UMLS
• Runtime Performance and Technical Considerations