Title: Tensor Methods for Nonlinear Matrix Completion,
Abstract: In the low rank matrix completion (LRMC) problem, the low rank assumption
means that the columns (or rows) of the matrix to be completed are points on a
low-dimensional linear algebraic variety. This paper extends this thinking to
cases where the columns are points on a low-dimensional nonlinear algebraic
variety, a problem we call Low Algebraic Dimension Matrix Completion (LADMC).
Matrices whose columns belong to a union of subspaces (UoS) are an important
special case. We propose a LADMC algorithm that leverages existing LRMC methods
on a tensorized representation of the data. For example, a second-order
tensorization representation is formed by taking the outer product of each
column with itself, and we consider higher order tensorizations as well. This
approach will succeed in many cases where traditional LRMC is guaranteed to
fail because the data are low-rank in the tensorized representation but not in
the original representation. We also provide a formal mathematical
justification for the success of our method. In particular, we show bounds of
the rank of these data in the tensorized representation, and we prove sampling
requirements to guarantee uniqueness of the solution. Interestingly, the
sampling requirements of our LADMC algorithm nearly match the information
theoretic lower bounds for matrix completion under a UoS model. We also provide
experimental results showing that the new approach significantly outperforms
existing state-of-the-art methods for matrix completion in many situations.