用稀疏采样和图模型完成高秩张量分解，提升推荐系统数据缺失时的预测能力。
Graphical model for factorization and completion of relatively high rank tensors by sparse sampling
- 基于随机图结构设计稀疏采样，建模高秩张量的因子分解。
- 在稠密但非全连通图下，实现接近贝叶斯最优的推理性能。
- 提出新方法避免高斯假设失效，适用于社交网络等缺失数据场景。
我们研究基于对相对高秩张量成分的稀疏测量进行张量分解的问题。测量设计使得潜在交互图结构为随机图，该设置在大量数据缺失时尤为适用，如社交网络服务中广泛使用的高秩矩阵补全问题。为获得理论洞察，我们考虑在高维极限（称为稠密极限）下的统计推断，此时图既大又稠密但非完全连接。我们构建了消息传递算法，并在特定情形下的贝叶斯最优教师-学生设定中进行了测试。此外，我们发展了一种基于累积量展开的复制理论，用于分析稠密极限中统计推断的性能，该方法可避免在全连接系统中失效的高斯假设盲用。
原文摘要 · Abstract (English)
We consider tensor factorizations based on sparse measurements of the components of relatively high rank tensors. 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 completion of relatively high rank matrices for 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 in some specific cases. We also develop a replica theory to examine the performance of statistical inference in the dense limit based on a cumulant expansion. The latter approach allows one to avoid blind usage of Gaussian ansatz which fails in some fully connected systems.
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