Algorithms for Matrix Estimation

在许多实际应用中，必须根据观察或测量的数据进行预测。在大多数情况下，测量的数据可以表示为具有多个缺失条目的大的m×n矩阵Z，并且问题变成了基于观察到的条目来“补全”该矩阵。这种矩阵估计问题的一个例子是推荐系统，其中基于用户过去偏好的部分知识来向用户推荐项目。通常，这可以通过应用低等级因子模型来实现。在数字信号处理的研究领域中，矩阵估计的目标是研究实现矩阵因子的这种低阶表示的最有效方法，并以尽可能低的误差恢复丢失的条目。在我在剑桥的第IIB部分项目中学习了矩阵估计，我理解了矩阵估计背后的原理，但也意识到了该技术的数学深度和与之相关的广泛可能性。矩阵估计在许多真实世界的应用中具有潜在的用途，因为它所需要的是两个实体之间的关联，从生物医学研究(基因疾病)到电影推荐系统(用户-电影关联)。在这些真实世界的应用中，通常有可用的数据(辅助信息)的先验知识，这些知识可以用来显著提高精度和降低复杂性。在我的博士学位期间，我想开发矩阵补全算法，以使用关于因子的不同类型的辅助信息。

There are many real-world applications in which predictions have to be made based on observed or measured data. In most cases, the measured data can be represented as a large m x n matrix Z with several missing entries and the problem becomes "completing" this matrix based on the observed entries. One such case of this matrix estimation problem is recommender systems, where items are recommended to users based on partial knowledge of their past preferences.Generally, this can be done by applying a low-rank factor model. A low-rank prediction of the original matrix can be obtained as the product of two factor matrices.Within the research area of digital signal processing, the goal of matrix estimation is to investigate the most effective ways of achieving this low-rank representation of the matrix factors and recover the missing entries with the lowest possible error.Having studied matrix estimation during my Part IIB Project in Cambridge, I have understood the principles behind matrix estimation but become aware of the mathematical depth of the technique and the broad range of possibilities associated with it. Matrix estimation has the potential to be useful in many real-world applications, as all it needs is associations between two entities, from biomedical research (gene-disease) to movie recommender systems (user-film associations).In these real-world applications, there is often previous knowledge of the data (side information) available which can be used to significantly improve accuracy as well as reducing complexity. During my PhD, I would like to develop matrix completion algorithms tailored to use different kinds of side information about the factors.