Matrix factorization (MF) is a simple collaborative filtering technique that achieves superior recommendation accuracy by decomposing the user-item rating matrix into user and item latent matrices. This approach relies on learning from user-item interactions, which may not effectively capture the underlying shared dependencies between users or items. Therefore, there is scope to explicitly capture shared dependencies to further improve recommendation accuracy and the interpretability of learning results by summarizing user-item interactions. Based on these insights, we propose "Hierarchical Matrix Factorization" (HMF), which incorporates clustering concepts to capture the hierarchy, where leaf nodes and other nodes correspond to users/items and clusters, respectively. Central to our approach, called hierarchical embeddings, is the additional decomposition of the user and item latent matrices (embeddings) into probabilistic connection matrices, which link the hierarchy, and a root cluster latent matrix. Thus, each node is represented by the weighted average of the embeddings of its parent clusters. The embeddings are differentiable, allowing simultaneous learning of interactions and clustering using a single gradient descent method. Furthermore, the obtained cluster-specific interactions naturally summarize user-item interactions and provide interpretability. Experimental results on rating and ranking predictions demonstrated the competitiveness of HMF over vanilla and hierarchical MF methods, especially its robustness in sparse interactions. Additionally, it was confirmed that the clustering integration of HMF has the potential for faster learning convergence and mitigation of overfitting compared to MF, and also provides interpretability through a cluster-centered case study.

翻译：矩阵分解（MF）是一种简单的协同过滤技术，通过将用户-物品评分矩阵分解为用户和物品的潜在矩阵，实现了优越的推荐精度。这种方法依赖于从用户-物品交互中学习，可能无法有效捕捉用户或物品之间潜在的共享依赖关系。因此，通过总结用户-物品交互来显式捕捉共享依赖关系，有望进一步提升推荐精度和学习结果的可解释性。基于这些见解，我们提出了“分层矩阵分解”（HMF），该方法融合了聚类概念以捕捉层次结构，其中叶节点对应于用户/物品，其他节点对应于聚类。该方法的核心理念——称为分层嵌入——在于将用户和物品潜在矩阵（嵌入）进一步分解为连接层次结构的概率连接矩阵以及根聚类潜在矩阵。因此，每个节点由其父聚类嵌入的加权平均值表示。这些嵌入是可微的，使得能够通过单一梯度下降法同时学习交互和聚类。此外，所获得的聚类特定交互自然地总结了用户-物品交互，并提供了可解释性。针对评分和排名预测的实验结果表明，HMF相较于普通MF和分层MF方法具有竞争力，尤其在稀疏交互场景下表现出更强的鲁棒性。此外，确认了与MF相比，HMF的聚类集成具有更快学习收敛和缓解过拟合的潜力，并通过以聚类为中心的案例研究提供了可解释性。