We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e., each data point is a solution to a system of polynomial equations. In this case, the original matrix is possibly high-rank, but it becomes low-rank after mapping each column to a higher dimensional space of monomial features. Many well-studied extensions of linear models, including affine subspaces and their union, can be described by a variety model. We study the sampling requirements for matrix completion under a variety model with a focus on a union of subspaces. We also propose an efficient matrix completion algorithm that minimizes a surrogate of the rank of the matrix of monomial features, which is able to recover synthetically generated data up to the predicted sampling complexity bounds. The proposed algorithm also outperforms standard low-rank matrix completion and subspace clustering techniques in experiments with real data.
