Algorithms for Matrix Estimation

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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.