Low-rank Matrix Completion using Alternating Minimization (1212.0467v1)

Abstract: Alternating minimization represents a widely applicable and empirically successful approach for finding low-rank matrices that best fit the given data. For example, for the problem of low-rank matrix completion, this method is believed to be one of the most accurate and efficient, and formed a major component of the winning entry in the Netflix Challenge. In the alternating minimization approach, the low-rank target matrix is written in a bi-linear form, i.e. $X = UV^\dag$; the algorithm then alternates between finding the best $U$ and the best $V$. Typically, each alternating step in isolation is convex and tractable. However the overall problem becomes non-convex and there has been almost no theoretical understanding of when this approach yields a good result. In this paper we present first theoretical analysis of the performance of alternating minimization for matrix completion, and the related problem of matrix sensing. For both these problems, celebrated recent results have shown that they become well-posed and tractable once certain (now standard) conditions are imposed on the problem. We show that alternating minimization also succeeds under similar conditions. Moreover, compared to existing results, our paper shows that alternating minimization guarantees faster (in particular, geometric) convergence to the true matrix, while allowing a simpler analysis.

Low-Rank Matrix Completion Using Alternating Minimization

The paper presents an in-depth analysis of alternating minimization for solving the low-rank matrix completion problem, as well as the related problem of matrix sensing. Traditionally, alternating minimization (AltMin) has been widely utilized in various data analysis contexts due to its empirical success. However, there has been scant theoretical backing for its efficiency and accuracy. This work bridges that gap by providing rigorous theoretical guarantees for AltMin, focusing on convergence properties and requirements.

Problem Statement

Low-rank matrix completion involves determining a low-rank matrix from a subset of its observed entries. This problem is crucial for applications such as recommender systems, where the task is to predict missing user-item ratings based on a sparse set of known ratings. Formally, given an $m \times n$ matrix $X$ of rank $k$, the objective is to fill in the missing entries given a set $\Omega$ of observed elements.

Alternating Minimization Approach

In the AltMin approach, the target low-rank matrix $X$ is expressed as a product of two smaller matrices $U \in \mathbb{R}^{m \times k}$ and $V \in \mathbb{R}^{n \times k}$, such that $X = UV^\dag$. The algorithm alternates between optimizing $U$ while keeping $V$ fixed, and vice versa. Each of these steps is a convex problem; however, the overall formulation is non-convex. Despite this non-convexity, AltMin is known for its computational efficiency and the ability to leverage sparsity and distributed computing.

Theoretical Contributions

The main theoretical contribution of the paper is the establishment of conditions under which the AltMin guarantees geometric convergence to the true low-rank solution. The key results can be summarized as follows:

1. Matrix Sensing: For matrix sensing, where the goal is to recover a matrix from linear measurements, the paper shows that if these measurements satisfy the Restricted Isometry Property (RIP), AltMin converges geometrically. Specifically, if the RIP constant $\delta_{2k}$ of the measurement operator $$is sufficiently small, the iterates of AltMin approach the true matrix exponentially fast.</li> <li><strong>Matrix Completion:</strong> For matrix completion, under the common assumptions of matrix incoherence and random sampling, the authors demonstrate that AltMin can recover the true low-rank matrix efficiently. They prove that the sampling complexity depends on the condition number of the matrix and the rank$$k$, with a tighter requirement for the RIP constant due to the non-independence of the matrix elements.
2. Computational Efficiency: Compared to convex relaxation methods, which require computing the singular value decomposition (SVD) at each step and thus have higher computational demands, AltMin requires only solving least squares problems. This leads to significant computational savings.

Stagewise Alternating Minimization

To address the dependence of convergence on the condition number of the matrix, the authors propose a stagewise variant of AltMin (Stage-AltMin). This variant incrementally increases the rank of the approximation, refining the previous estimate at each stage. This method achieves near-optimal sample complexity while maintaining computational efficiency.

Implications and Future Directions

The rigorous analysis provided by the authors for AltMin brings it on par with other theoretically grounded methods like nuclear norm minimization. The practical implications are profound, providing a clear alternative that is computationally efficient for large-scale data sets.

The results also open up several avenues for future research. Potential directions include:

• Extending the analysis to other structured low-rank reconstruction problems, such as tensor completion.
• Developing enhanced initialization techniques that can further reduce sample complexity.
• Investigating the implications of AltMin variants in distributed and parallel computing environments.

In conclusion, this paper substantially contributes to the theoretical understanding of alternating minimization for low-rank matrix completion and sensing. The demonstrated geometric convergence under reasonable conditions, and the introduction of a stagewise variation, present exciting opportunities for the application of this method in various domains of large-scale data analysis.