CMF is a technique for simultaneously learning low-rank representations based on a collection of matrices with shared entities. A typical example is the joint modeling of user-item, item-property, and user-feature matrices in a recommender system. The key idea in CMF is that the embeddings are shared across the matrices, which enables transferring information between them. The existing solutions, however, break down when the individual matrices have low-rank structure not shared with others. In this work we present a novel CMF solution that allows each of the matrices to have a separate low-rank structure that is independent of the other matrices, as well as structures that are shared only by a subset of them. We compare MAP and variational Bayesian solutions based on alternating optimization algorithms and show that the model automatically infers the nature of each factor using group-wise sparsity. Our approach supports in a principled way continuous, binary and count observations and is efficient for sparse matrices involving missing data. We illustrate the solution on a number of examples, focusing in particular on an interesting use-case of augmented multi-view learning.

本研究提出了一种基于协作矩阵分解(CMF)的新算法，允许每个矩阵具有单独的低秩结构，并支持仅某些矩阵之间共享的结构。提出的方法采用交替优化算法，比较了MAP和变分贝叶斯解，并展示了通过组稀疏自动推断因子的本质，支持连续、二进制和计数观测，并支持缺失数据的稀疏矩阵。该算法应用于增强型多视角学习等多个应用案例中。

集体矩阵分解中的群稀疏嵌入