用线性关系补全矩阵，解决推荐系统与图嵌入问题
Low rank matrix completion and realization of graphs: results and problems
- 通过线性约束补全低秩矩阵，替代传统缺失值预测
- 可应用于图在曲面上的嵌入（含旋转系统与模2情形）
- 适合研究推荐系统、图论与几何建模的读者
Netflix问题（机器学习中的经典问题）提出：给定一个评分矩阵，其中条目$(i,j)$表示用户$i$对电影$j$的评分（若用户看过该电影），否则为空缺。目标是预测剩余空缺条目以提供精准推荐，且使补全后的矩阵秩最小。本文综述更一般的问题：不直接已知具体矩阵元素，而是已知这些元素间的线性关系。我们描述此类结果在图嵌入于曲面（更精确地说，具有旋转系统或模2意义下的嵌入）方面的应用。
原文摘要 · Abstract (English)
The Netflix problem (from machine learning) asks the following. Given a ratings matrix in which each entry $(i,j)$ represents the rating of movie $j$ by customer $i$, if customer $i$ has watched movie $j$, and is otherwise missing, we would like to predict the remaining entries in order to make good recommendations to customers on what to watch next. The remaining entries are predicted so as to minimize the {\it rank} of the completed matrix. In this survey we study a more general problem, in which instead of knowing specific matrix elements, we know linear relations on such elements. We describe applications of these results to embeddings of graphs in surfaces (more precisely, embeddings with rotation systems, and embeddings modulo 2).
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