# SET: an algorithm for consistent matrix completion

## TL;DR

The paper introduces the SET algorithm for consistent matrix completion, which effectively finds low-rank matrices matching observed entries by evolving and transferring subspaces, demonstrating strong empirical results.

## Contribution

It presents a novel subspace evolution and transfer algorithm that addresses barriers in the search process for consistent low-rank matrix completion.

## Key findings

- Excellent empirical performance on very low-rank matrices
- Effective barrier detection and transfer mechanisms
- Improved convergence in matrix completion tasks

## Abstract

A new algorithm, termed subspace evolution and transfer (SET), is proposed for solving the consistent matrix completion problem. In this setting, one is given a subset of the entries of a low-rank matrix, and asked to find one low-rank matrix consistent with the given observations. We show that this problem can be solved by searching for a column space that matches the observations. The corresponding algorithm consists of two parts -- subspace evolution and subspace transfer. In the evolution part, we use a line search procedure to refine the column space. However, line search is not guaranteed to converge, as there may exist barriers along the search path that prevent the algorithm from reaching a global optimum. To address this problem, in the transfer part, we design mechanisms to detect barriers and transfer the estimated column space from one side of the barrier to the another. The SET algorithm exhibits excellent empirical performance for very low-rank matrices.