The Power of Convex Relaxation: Near-Optimal Matrix Completion (0903.1476v1)

Abstract: This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix Prize and other similar questions in collaborative filtering. In general, accurate recovery of a matrix from a small number of entries is impossible; but the knowledge that the unknown matrix has low rank radically changes this premise, making the search for solutions meaningful. This paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever (the information theoretic limit). More importantly, the paper shows that, under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit (up to logarithmic factors). This convex program simply finds, among all matrices consistent with the observed entries, that with minimum nuclear norm. As an example, we show that on the order of nr log(n) samples are needed to recover a random n x n matrix of rank r by any method, and to be sure, nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog(n).

Overview of "The Power of Convex Relaxation: Near-Optimal Matrix Completion"

This paper by Emmanuel J. Candès and Terence Tao rigorously examines the problem of matrix completion, especially in the context of recovering a low-rank matrix from a limited set of its entries. The authors present novel conditions and results that considerably improve the theoretical understanding and practical feasibility of matrix completion using techniques from convex relaxation.

Unlike previous works that provided sub-optimal conditions for exact recovery, this paper introduces near-optimal conditions under which matrix completion can be achieved with a number of entries close to the theoretical minimum necessary to capture the low-rank structure. Specifically, the authors demonstrate that under certain incoherence conditions on the singular vectors of the matrix, recovery through nuclear norm minimization succeeds with high probability when the number of observed entries is proportional to $nr \log n$, where $n$ is the matrix dimension, and $r$ is its rank.

Main Contributions

Information Theoretic Limit and Convex Optimization:

• The paper establishes that the optimal number of entries required for matrix completion is related to the rank $r$ and the dimension $n$, introducing the notion of the information-theoretic limit. The significant finding is that nuclear norm minimization can achieve exact recovery even when the sampling rate is only a logarithmic factor above this limit.

Incoherence Conditions:

• Candès and Tao introduce two key notions of incoherence. The strong incoherence property with parameter $\mu$ and another, slightly different, incoherence condition previously described in the literature. They show that if $\mu$ is small, the number of required samples scales as $nr \text{polylog}(n)$.

Recovery Guarantees:

• Using a dual certificate approach, the authors provide sufficient conditions that guarantee nuclear norm minimization succeeds. They construct a candidate dual matrix and derive conditions under which this matrix certifies the optimality of the solution.

Numerical Results and Implications

The paper provides robust theoretical results supported by significant technical derivations, including:

1. Moment Bound Analysis: The authors use moment bounds to estimate the spectral norm of certain random matrices arising in their analysis. This moment method is critical in providing tight bounds on the probability that their proposed conditions are satisfied.
2. Strong Incoherence Property: They show through a detailed combinatorial argument that the strong incoherence property with small $\mu$ is sufficient for near-optimal matrix completion. This finding significantly broadens the set of matrices that can be recovered via convex relaxation.
3. Coupling with Random Matrix Theory: The paper extensively uses techniques from random matrix theory, including concentration inequalities and spectral norm estimates, to justify the presented results.

Practical and Theoretical Implications

Practical Implications:

• Matrix Completion in Recommender Systems:

Applications such as the Netflix Prize and other recommender systems can significantly benefit from these results. The ability to accurately predict user preferences from a sparse set of data points is crucial for improving recommendation accuracy and user satisfaction.

• Computer Vision and Remote Sensing:

Matrix completion techniques find applications in computer vision tasks like inpainting missing parts of images or videos and in remote sensing where full data matrices need to be inferred from limited observations.

Theoretical Implications:

• Compressed Sensing Analogy:

The results draw a parallel to compressed sensing, showing that a convex relaxation approach can achieve nearly optimal recovery in different contexts, thus expanding the theoretical foundation of optimization-based recovery techniques.

• Future Research Directions:

Candès and Tao's work opens avenues for further research on robust matrix completion, where entries might be corrupted by noise. Extending these results to handle noisy observations is a critical next step.

Conclusion

Candès and Tao provide a rigorous and deep examination of matrix completion through convex relaxation. By introducing and leveraging the strong incoherence property and using advanced tools from random matrix theory, they significantly improve the understanding of the fundamental limits of matrix completion. Their work lays the groundwork for both theoretical advancements and practical implementations in various fields requiring the recovery of low-rank matrices from incomplete data.