Abstract

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 s tion 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, as well as a rich class of nonlinear quadratic and higher degree curves and surfaces. We study the sampling requirements for matrix completion under a variety model with a focus on a union of affine subspaces. We also propose an efficient matrix completion algorithm that minimizes a convex or non-convex surrogate of the rank of the matrix of monomial features, using the wellknown “kernel trick” to avoid working directly with the high-dimensional monomial matrix. We show the proposed algorithm is able to recover synthetically generated data up to the predicted sampling complexity bounds, and outperforms standard low rank matrix completion and subspace clustering algorithms in experiments with real data.