Matrix Completion from a Few Entries (0901.3150v4)

Abstract: Let M be a random (alpha n) x n matrix of rank r<<n, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm that reconstructs M from |E| = O(rn) observed entries with relative root mean square error RMSE <= C(rn/|E|)^0.5. Further, if r=O(1), M can be reconstructed exactly from |E| = O(n log(n)) entries. These results apply beyond random matrices to general low-rank incoherent matrices. This settles (in the case of bounded rank) a question left open by Candes and Recht and improves over the guarantees for their reconstruction algorithm. The complexity of our algorithm is O(|E|r log(n)), which opens the way to its use for massive data sets. In the process of proving these statements, we obtain a generalization of a celebrated result by Friedman-Kahn-Szemeredi and Feige-Ofek on the spectrum of sparse random matrices.

• The paper introduces a spectral matrix completion algorithm that reconstructs low-rank matrices from partial entries with low relative RMSE and exact recovery conditions.
• The method employs a three-step process of trimming, SVD projection, and cleaning, reducing computational complexity to O(|E|r log n) for large-scale data sets.
• Implications extend to recommendation systems, image processing, and machine learning, offering scalable recovery techniques for incomplete data.

Efficient Matrix Completion from a Few Entries

Introduction

The paper "Matrix Completion from a Few Entries" by Raghunandan H. Keshavan, Andrea Montanari, and Sewoong Oh addresses the problem of reconstructing a low-rank matrix from a small subset of its entries. This issue is highly relevant in various fields such as collaborative filtering, where one aims to predict unknown ratings in a user-item context. The authors propose an efficient algorithm that achieves this reconstruction with a small number of observed entries, advancing beyond the guarantees of existing methods.

Problem Formulation and Model

The problem is formally defined for a matrix $M$ of dimensions $n\alpha \times n$ and rank $r$. Only a uniformly random subset $E$ of the matrix's entries is observed. The goal is to accurately reconstruct $M$ from these partial observations.

The matrix $M$ is assumed to factorize as $M = U \Sigma V^T$. Here, $U$ and $V$ are matrices of dimensions $n\alpha \times r$ and $n \times r$ respectively, and $\Sigma$ is a diagonal matrix of dimensions $r \times r$. The entries of $U$ and $V$ are assumed to be sufficiently unstructured, satisfying an incoherence condition.

Algorithm

The proposed algorithm, named Spectral Matrix Completion, consists of multiple steps:

1. Trimming: High-degree rows and columns are systematically zeroed out to remove noise components.
2. Projection: The trimmed matrix undergoes Singular Value Decomposition (SVD), and the rank-$r$ approximation is computed.
3. Cleaning: The rank-$r$ approximation is refined to minimize residual errors through local optimization.

The algorithm performs these steps efficiently with a computational complexity of $O(|E|r \log n)$, making it suitable for large-scale data sets.

Theoretical Results

The authors provide stringent theoretical guarantees for their method. In particular, they prove two main results:

1. Relative Root Mean Square Error (RMSE):

   $$\text{RMSE} \leq C(\alpha) \left( \frac{nr}{|E|} \right)^{1/2}$$

This result indicates that the algorithm achieves arbitrarily small RMSE from $O(nr)$ observed entries.

1. Exact Reconstruction: If $$|E| \geq C'(\alpha) n r \sqrt{\alpha} \max(\log n, r \sqrt{\alpha})$$, and the matrices $U$ and $V$ meet the incoherence conditions, the algorithm will reconstruct $M$ exactly with high probability.

The theoretical analysis is backed by understanding the spectral properties of the observed and trimmed matrices, similarity to properties of random graphs, and optimization over Grassmannian manifolds.

Implications and Future Directions

The practical implications of this work are significant, particularly for applications in recommendation systems, clustering, image processing, and machine learning where low-rank approximations are prevalent. The reduction of computational complexity without sacrificing accuracy makes this approach highly desirable for large-scale data sets.

Future work can focus on refining the exact conditions and thresholds necessary for exact matrix completion, possibly extending the analysis to more structured or deterministic sets $E$. Additionally, non-incoherent matrices and other low-rank matrix models could be considered to broaden the applicability of these methods.

Conclusion

The paper introduces an efficient and theoretically sound algorithm for matrix completion, providing strong performance guarantees with practical efficiencies. This advancement holds promise for numerous applications that rely on partial data, ensuring both accuracy and scalability.