The use of graph convolution in the development of recommender system algorithms has recently achieved state-of-the-art results in the collaborative filtering task (CF). While it has been demonstrated that the graph convolution operation is connected to a filtering operation on the graph spectral domain, the theoretical rationale for why this leads to higher performance on the collaborative filtering problem remains unknown. The presented work makes two contributions. First, we investigate the effect of using graph convolution throughout the user and item representation learning processes, demonstrating how the latent features learned are pushed from the filtering operation into the subspace spanned by the eigenvectors associated with the highest eigenvalues of the normalised adjacency matrix, and how vectors lying on this subspace are the optimal solutions for an objective function related to the sum of the prediction function over the training data. Then, we present an approach that directly leverages the eigenvectors to emulate the solution obtained through graph convolution, eliminating the requirement for a time-consuming gradient descent training procedure while also delivering higher performance on three real-world datasets.

翻译：图卷积在推荐系统算法开发中的应用近期在协同过滤任务（CF）中取得了最先进的结果。尽管已有研究表明图卷积操作与图谱域上的滤波操作相关联，但该操作为何能提升协同过滤问题性能的理论依据仍不明确。本研究包含两项贡献：首先，我们探究了在用户与物品表示学习过程中使用图卷积的影响，证明了学习到的潜在特征如何被滤波操作推向归一化邻接矩阵最大特征值对应特征向量张成的子空间，以及位于该子空间上的向量如何成为与训练数据预测函数求和相关目标函数的最优解。在此基础上，我们提出一种直接利用特征向量模拟图卷积所得解的方法，不仅消除了耗时的梯度下降训练过程，还在三个真实世界数据集上实现了更优性能。