We consider tensor factorizations based on sparse measurements of the tensor components. The measurements are designed in a way that the underlying graph of interactions is a random graph. The setup will be useful in cases where a substantial amount of data is missing, as in recommendation systems heavily used in social network services. In order to obtain theoretical insights on the setup, we consider statistical inference of the tensor factorization in a high dimensional limit, which we call as dense limit, where the graphs are large and dense but not fully connected. We build message-passing algorithms and test them in a Bayes optimal teacher-student setting. We also develop a replica theory, which becomes exact in the dense limit,to examine the performance of statistical inference.

翻译：我们考虑基于张量分量稀疏测量的张量分解方法。测量设计使得底层交互图呈现随机图结构。该框架适用于数据大量缺失的场景，例如社交网络服务中广泛使用的推荐系统。为获得该框架的理论洞见，我们在高维极限（称为稠密极限）下研究张量分解的统计推断问题，此时交互图规模庞大且稠密但非全连接。我们构建了消息传递算法，并在贝叶斯最优的师生设置中进行测试。同时发展了复制理论（该理论在稠密极限下精确成立）以检验统计推断的性能。