Latent variable collaborative filtering methods have been a standard approach to modelling user-click interactions due to their simplicity and effectiveness. However, there is limited work on analyzing the mathematical properties of these methods in particular on preventing the overfitting towards the identity, and such methods typically utilize loss functions that overlook the geometry between items. In this work, we introduce a notion of generalization gap in collaborative filtering and analyze this with respect to latent collaborative filtering models. We present a geometric upper bound that gives rise to loss functions, and a way to meaningfully utilize the geometry of item-metadata to improve recommendations. We show how these losses can be minimized and gives the recipe to a new latent collaborative filtering algorithm, which we refer to as GeoCF, due to the geometric nature of our results. We then show experimentally that our proposed GeoCF algorithm can outperform other all existing methods on the Movielens20M and Netflix datasets, as well as two large-scale internal datasets. In summary, our work proposes a theoretically sound method which paves a way to better understand generalization of collaborative filtering at large.

翻译：潜在变量协同过滤方法因其简洁性和有效性，已成为建模用户点击交互的标准方法。然而，目前对这些方法数学特性的分析研究较为有限，特别是在防止向恒等映射过拟合方面存在不足，且此类方法通常采用的损失函数忽略了物品间的几何关系。本研究引入协同过滤中泛化间隙的概念，并针对潜在协同过滤模型进行分析。我们提出了一个能推导出损失函数的几何上界，以及一种有效利用物品元数据几何结构以改进推荐的方法。我们展示了如何最小化这些损失，并构建了一种新的潜在协同过滤算法——鉴于其几何特性，我们将其命名为GeoCF。实验结果表明，我们提出的GeoCF算法在Movielens20M和Netflix数据集以及两个大规模内部数据集上均优于现有所有方法。总之，本研究提出了一种理论完备的方法，为深入理解协同过滤的泛化特性开辟了新途径。