Subspace Evolution and Transfer (SET) for Low-Rank Matrix Completion

TL;DR

The paper introduces SET, a novel algorithm for low-rank matrix completion that uses subspace evolution and transfer on the Grassmann manifold, demonstrating strong empirical results across various sampling rates.

Contribution

The paper proposes a new subspace evolution and transfer algorithm for matrix completion, incorporating barrier detection and transfer mechanisms on the Grassmann manifold.

Findings

Excellent empirical performance across different sampling regimes

Effective barrier detection and transfer mechanism

Improved convergence in low-rank matrix completion

Abstract

We describe a new algorithm, termed subspace evolution and transfer (SET), for solving low-rank matrix completion problems. The algorithm takes as its input a subset of entries of a low-rank matrix, and outputs one low-rank matrix consistent with the given observations. The completion task is accomplished by searching for a column space on the Grassmann manifold that matches the incomplete observations. The SET algorithm consists of two parts -- subspace evolution and subspace transfer. In the evolution part, we use a gradient descent method on the Grassmann manifold to refine our estimate of the column space. Since the gradient descent algorithm is not guaranteed to converge, due to the existence of barriers along the search path, we design a new mechanism for detecting barriers and transferring the estimated column space across the barriers. This mechanism constitutes the core of the transfer step of the algorithm. The SET algorithm exhibits excellent empirical performance for both high and low sampling rate regimes.