Matrix completion with columns in union and sums of subspaces

Motivated by missing data recovery for power system monitoring, we study the problem of matrix completion when each column belongs to either one of a few low-dimensional subspaces or the sum of a subset of subspaces. This novel model "union and sums of subspaces" is a generalization of the model of a union of subspaces (UoS) studied under subspace clustering. It characterizes the low-dimensional nonlinear structure in high-dimensional data points. We propose a convex-optimization-based method to recover missing entries under this model and provide its theoretical guarantee. Numerical experiment on power system data is conducted to verify the effectiveness of the method.