Low-rank Matrix Recovery from Errors and Erasures (1104.0354v2)

Abstract: This paper considers the recovery of a low-rank matrix from an observed version that simultaneously contains both (a) erasures: most entries are not observed, and (b) errors: values at a constant fraction of (unknown) locations are arbitrarily corrupted. We provide a new unified performance guarantee on when the natural convex relaxation of minimizing rank plus support succeeds in exact recovery. Our result allows for the simultaneous presence of random and deterministic components in both the error and erasure patterns. On the one hand, corollaries obtained by specializing this one single result in different ways recover (up to poly-log factors) all the existing works in matrix completion, and sparse and low-rank matrix recovery. On the other hand, our results also provide the first guarantees for (a) recovery when we observe a vanishing fraction of entries of a corrupted matrix, and (b) deterministic matrix completion.

• The paper introduces a unified theoretical framework providing guarantees for exact recovery of low-rank matrices from data containing both missing entries (erasures) and corrupted entries (errors).
• Key theorems establish improved conditions for recovery even with a vanishing fraction of observations, random error signs, or worst-case adversarial error patterns.
• The methodology uses a convex program minimizing nuclear and L1 norms, validated by constructing dual certificates via a combined Golfing Scheme and least squares approach for mixed error types.

Low-rank Matrix Recovery from Errors and Erasures: A Technical Analysis

The paper "Low-rank Matrix Recovery from Errors and Erasures" addresses a fundamental problem in the field of data analysis and dimensionality reduction: the recovery of low-rank matrices from observed data that is incomplete and potentially corrupted. The authors introduce a theoretical framework for guaranteeing exact recovery of these matrices under a unified model considering both erasures (missing entries) and errors (corrupted entries). This work is particularly relevant for applications such as Principal Component Analysis (PCA), collaborative filtering, and spectral clustering, where the underlying data matrix is often only partially observable and corrupted.

Core Contributions

The primary contribution of the paper is the establishment of a unified performance guarantee for the convex relaxation strategy of minimizing rank plus support. This guarantee provides the conditions under which exact matrix recovery is achievable, being a novel result in that it simultaneously accounts for random and deterministic patterns in errors and erasures. Utilizing the framework of incoherence conditions, the authors demonstrate that recovery is possible even with a vanishing fraction of observed entries, which is significant for applications with sparse observations like collaborative filtering.

The authors present three main theorems:

1. Unified Guarantee (Theorem 1): This theorem provides conditions under which recovering the matrix is possible with high probability, even when entries are observed at random and corruption occurs at unknown locations. The conditions improve upon previous work by allowing for a smaller fraction of observed entries and addressing deterministic erasure contexts for the first time.
2. Improved Guarantee for Errors with Random Sign (Theorem 2): By assuming that the signs of the error matrix are random, the authors show that it is possible to recover the matrix even when almost all entries are corrupted. This result extends the probabilistic guarantees beyond previous findings, allowing for recovery with minimal observations.
3. Improved Deterministic Guarantee (Theorem 3): This theorem provides conditions for exact recovery under worst-case error patterns, improving upon previous deterministic results by allowing for larger sets of adversarial corruptions.

Methodological Approach

The recovery algorithm employed is a convex program utilizing nuclear norm minimization as a surrogate for matrix rank and the $\ell_1$ norm for sparsity of errors. The authors validate the conditions necessary for exact recovery by constructing dual certificates using the Golfing Scheme method combined with least squares approaches, which innovatively accommodates the presence of both random and adversarial error patterns.

Numerical Results and Implications

The paper includes numerical experiments that demonstrate the robustness of the proposed recovery method as the matrix size increases, highlighting the relaxing conditions required for successful recovery. These findings align with theoretical predictions, substantiating the practical implications of the recovery strategy in real-world data applications.

The implications of this research are both theoretical and practical. It advances the understanding of when convex relaxation approaches can successfully recover matrices under combined random and deterministic corruptions. Practically, this contributes to fields relying on large-scale data analysis, enabling improved handling of data incompleteness and corruption.

Future Directions

The paper suggests several potential avenues for future research. These include exploring more efficient computational methods for large-scale implementations, as well as extending the theoretical framework to handle structured noise patterns beyond random signs and adversarial blocks. Further investigation into applications in collaborative filtering, computer vision, and bioinformatics may further realize the impact of this work.

Conclusion

Overall, this paper contributes significantly to the matrix recovery literature by expanding the theoretical boundaries of when recovery is feasible under challenging conditions. The rigorous approach to defining recovery guarantees sets a foundation for subsequent research to build upon and adapt these techniques to various domains where data integrity is a critical concern.