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Low algebraic dimension matrix completion

Low rank matrix completion (LRMC) has received tremendous attention in recent years. The low rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear variety. This paper extends this thinking to cases where the columns are points on low-dimensional nonlinear algebraic varieties. While others have recently studied matrix completion in such settings, existing results focus mainly on algorithms and experiments, without supporting theory. This paper proposes a new approach to what we call Low Algebraic-Dimension Matrix Completion (LADMC). We propose a new LADMC algorithm that leverages existing LRMC methods on a tensorized representation of the data. We also provide a formal mathematical justification for the success of our method. In particular, the new algorithm can succeed in many cases where traditional LRMC is guaranteed to fail. We also provide experimental results showing that the new approach significantly outperforms existing state-of-the-art methods for matrix completion in many situations.