A Gradient Descent Algorithm on the Grassman Manifold for Matrix Completion

TL;DR

This paper introduces OptSpace, an efficient algorithm combining SVD and manifold optimization for low-rank matrix completion, demonstrating high accuracy from minimal data and robustness to noise.

Contribution

The paper presents a novel algorithm that effectively reconstructs low-rank matrices from limited entries using a combination of SVD and Grassman manifold optimization.

Findings

Accurately reconstructs low-rank matrices from few entries.

Demonstrates robustness to noisy data.

Performs well on real collaborative filtering datasets.

Abstract

We consider the problem of reconstructing a low-rank matrix from a small subset of its entries. In this paper, we describe the implementation of an efficient algorithm called OptSpace, based on singular value decomposition followed by local manifold optimization, for solving the low-rank matrix completion problem. It has been shown that if the number of revealed entries is large enough, the output of singular value decomposition gives a good estimate for the original matrix, so that local optimization reconstructs the correct matrix with high probability. We present numerical results which show that this algorithm can reconstruct the low rank matrix exactly from a very small subset of its entries. We further study the robustness of the algorithm with respect to noise, and its performance on actual collaborative filtering datasets.