Matrix Completion With Noise (0903.3131v1)

Published 18 Mar 2009 in cs.IT and math.IT

Abstract: On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest form, the problem is to recover a matrix from a small sample of its entries, and comes up in many areas of science and engineering including collaborative filtering, machine learning, control, remote sensing, and computer vision to name a few. This paper surveys the novel literature on matrix completion, which shows that under some suitable conditions, one can recover an unknown low-rank matrix from a nearly minimal set of entries by solving a simple convex optimization problem, namely, nuclear-norm minimization subject to data constraints. Further, this paper introduces novel results showing that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise. A typical result is that one can recover an unknown n x n matrix of low rank r from just about nr log² n noisy samples with an error which is proportional to the noise level. We present numerical results which complement our quantitative analysis and show that, in practice, nuclear norm minimization accurately fills in the many missing entries of large low-rank matrices from just a few noisy samples. Some analogies between matrix completion and compressed sensing are discussed throughout.

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Authors (2)

Yaniv Plan (33 papers)
Emmanuel J. Candès (60 papers)

Citations (1,747)

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Summary

Matrix Completion with Noise: An Analytical Essay

This essay provides an insightful overview of the paper titled "Matrix Completion with Noise" by Emmanuel J. Candès and Yaniv Plan, from Applied and Computational Mathematics, Caltech. The paper discusses the problem of recovering a data matrix from incomplete and corrupted information, specifically focusing on matrix completion under noisy conditions.

Overview

Matrix completion is the task of inferring missing entries of a partially observed matrix under the assumption that the matrix is of low rank. This problem is prevalent in various fields such as collaborative filtering, machine learning, control, remote sensing, and computer vision. The authors present a compelling case that low-rank matrix completion can be successfully conducted even when observed entries are noisily sampled.

Core Contributions

Exact Matrix Completion: The paper explores the theoretical underpinnings of matrix completion. A significant portion is dedicated to exploring the conditions under which an unknown matrix $M \in \mathbb{R}^{n \times n}$ can be exactly recovered from a small subset of its entries. The recovery is framed as a convex optimization problem, specifically nuclear-norm minimization. The paper reveals that under suitable incoherence conditions, it is possible to recover a low-rank matrix from approximately $nr \log^2 n$ noisy samples with an error proportional to the noise level.
Noisy Matrix Completion: The paper extends the matrix completion problem to scenarios where the observed entries are corrupted by noise. The authors rigorously prove that the nuclear-norm minimization method remains robust even when the observations are noisy. The optimization problem in the presence of noise is expressed as:

$\text{minimize} \quad \|X\|_* \quad \text{subject to} \quad \|P(X - Y)\|_F \leq \delta$

Here, $Y$ represents the noisy observations, and $\delta$ quantifies the noise level.

Dual Certificates: A critical theoretical contribution is the introduction of the concept of dual certificates. Dual certificates are employed to demonstrate that the nuclear-norm minimization problem has a unique solution under certain conditions. The authors prove that if there exists a dual certificate $\Lambda$ with $\|P(\Lambda)\| \leq 1/2$ and if $P_{\Omega}P_{\Omega}P_{\Omega} \succeq \frac{p}{2} I$ , then the matrix completion problem is both exact and stable in the presence of noise.
Numerical Experiments: Empirical validation is conducted through extensive numerical experiments, emphasizing the practical applicability of the theoretical results. The experiments show that the proposed method can denoise observed entries and accurately predict unobserved entries. Importantly, the experiments underline the method's effectiveness even when only a fraction of the matrix entries are observed.

Implications

The theoretical and empirical results presented have significant practical and theoretical implications:

Practical Applications:

This research demonstrates the feasibility of accurately recovering low-rank matrices in real-world applications, ranging from recommendation systems (e.g., Netflix, Amazon) to system identification in control systems and sensor networks.

Theoretical Advances:

From a theoretical standpoint, the results contribute to the fields of signal processing and compressed sensing. The introduction of dual certificates provides a new tool for establishing the uniqueness of solutions in matrix completion problems.

Future Developments:

Future research could explore more efficient algorithms for solving the nuclear-norm minimization problem, particularly for large-scale matrices. Additionally, further investigation is warranted into the tightness of the theoretical bounds and potential improvements in the presence of stochastic noise.

Conclusion

The paper "Matrix Completion with Noise" by Candès and Plan provides robust theoretical foundations and practical methodologies for matrix completion in the presence of noise. The insights gained from this research are likely to influence future developments in various applied and computational mathematics fields, especially those dealing with incomplete and noisy data. The combination of theoretical rigor and empirical validation ensures that the findings are both scientifically and practically relevant.

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