IGMC，Inductive graph-based matrix completion，基于归纳图的矩阵完成

基于归纳图的矩阵完成 (IGMC，Inductive graph-based matrix completion)

一、动机
用于矩阵补全的矩阵分解方法本质上是转导性的（transductive）。当矩阵发生改变时，通常需要再次进行训练才能得到新的嵌入。

①为什么矩阵分解可以用于矩阵补全任务？

图1 - 传统矩阵分解在评分矩阵上的应用

矩阵分解（Matrix Factorization）是指用 M’= A×B 来近似矩阵M，那么 A×B 的元素就可以用于估计M中对应不可见位置的元素值，而A×B可以看做是M的分解，所以称作矩阵分解。

矩阵补全（Matrix Completion）目的是为了估计矩阵中缺失的部分（不可观察的部分），可以看做是用矩阵X近似矩阵M，然后用X中的元素作为矩阵M中不可观察部分的元素的估计。

用M'=A×B来近似M，再用M'上的元素值来填充M上的缺失值，达到预测效果进而补全矩阵。

②Transductive Learning（转导性学习）和Inductive Learning（归纳性学习）的区别？

利用可交换矩阵层（Exchangeable Matrix Layers）实现矩阵补全的方法尽管是具有归纳性的，但是其会将整个矩阵作为输入并输出一个重构的矩阵。

为了减少对辅助信息的依赖，2018年的《Deep Model of Interactions Across Sets, 跨集合交互的深度模型》提出了可交换矩阵层（Exchangeable Matrix Layers），它对评分矩阵应用置换等变运算来重建缺失条目，得到的模型是具有归纳性的，并且不依赖于辅助信息。然而，这些模型将整个评分矩阵R作为输入，输出另一个重构的矩阵，这对于扩展到具有数百万用户/项目的实际数据集中来说是非常困难的。

About

IGMC is an inductivematrix completion model based on graph neural networks without using any side information. Traditional matrix factorization approaches factorize the (rating) matrix into the product of low-dimensional latent embeddings of rows (users) and columns (items), which are transductive since the learned embeddings cannot generalize to unseen rows/columns or to new matrices. To make matrix completion inductive, content (side information), such as user's age or movie's genre, has to be used previously. However, high-quality content is not always available, and can be hard to extract. Under the extreme setting where not any side information is available other than the matrix to complete, can we still learn an inductive matrix completion model? IGMC achieves this by training a graph neural network (GNN) based purely on local subgraphs around (user, item) pairs extracted from the bipartite graph formed by the rating matrix, and maps these subgraphs to their corresponding ratings. It does not rely on any global information specific to the rating matrix or the task, nor does it learn embeddings specific to the observed users/items. Thus, IGMC is a completely inductive model.

Since IGMC is inductive, it can generalize to users/items unseen during the training (given that their interactions exist), and can even transfer to new tasks. Our transfer learning experiments show that a model trained out of the MovieLens dataset can be directly used to predict Douban movie ratings and works surprisingly well. For more information, please check our paper:

M. Zhang and Y. Chen, Inductive Matrix Completion Based on Graph Neural Networks. [PDF]