Robust Low-rank Tensor Recovery: Models and Algorithms (1311.6182v1)

Abstract: Robust tensor recovery plays an instrumental role in robustifying tensor decompositions for multilinear data analysis against outliers, gross corruptions and missing values and has a diverse array of applications. In this paper, we study the problem of robust low-rank tensor recovery in a convex optimization framework, drawing upon recent advances in robust Principal Component Analysis and tensor completion. We propose tailored optimization algorithms with global convergence guarantees for solving both the constrained and the Lagrangian formulations of the problem. These algorithms are based on the highly efficient alternating direction augmented Lagrangian and accelerated proximal gradient methods. We also propose a nonconvex model that can often improve the recovery results from the convex models. We investigate the empirical recoverability properties of the convex and nonconvex formulations and compare the computational performance of the algorithms on simulated data. We demonstrate through a number of real applications the practical effectiveness of this convex optimization framework for robust low-rank tensor recovery.

• The paper presents a convex optimization framework utilizing ADAL and APG methods that guarantee global convergence for robust tensor recovery.
• It introduces a nonconvex model that leverages additional rank information to enhance recovery accuracy in the presence of outliers and corruptions.
• Empirical results demonstrate superior performance over traditional approaches in applications like computer vision, web data mining, and signal processing.

Robust Low-rank Tensor Recovery: Models and Algorithms

The paper "Robust Low-rank Tensor Recovery: Models and Algorithms" by Donald Goldfarb and Zhiwei Qin addresses a fundamental problem in multilinear data analysis, specifically focusing on robust tensor decomposition in the presence of outliers, gross corruptions, and missing data. This research proposes both convex and nonconvex models for recovering low-rank tensors, offering algorithms with global convergence guarantees, which have valuable applications across various domains including computer vision, web data mining, and signal processing.

Overview of Approaches

The researchers delve into developing a convex optimization framework to tackle the problem of robust low-rank tensor recovery. Building upon the foundations of robust Principal Component Analysis (PCA) and tensor completion, this framework utilizes tailored optimization algorithms based on alternating direction augmented Lagrangian (ADAL) and accelerated proximal gradient (APG) methods. Notably, these algorithms address both constrained and Lagrangian formulations of the recovery problem. Furthermore, the paper introduces a nonconvex model aimed at potentially enhancing recovery results compared to the convex models.

Key Contributions and Numerical Results

The pivotal contributions of the paper include:

1. Optimization Algorithms: The introduction of efficient ADAL-based algorithms allows for solving the constrained version of the robust tensor recovery problem, effectively handling the need for low-rank reconstruction in noisy data environments. These algorithms provide global convergence assurances, a non-trivial guarantee considering the inherent complexity of the problem.
2. Lagrangian Formulation with APG: The exploration of the Lagrangian formulation using APG addresses scenarios where strict consistency is relaxed, enabling practical trade-offs between precision and computational feasibility.
3. Nonconvex Model: By proposing a nonconvex model, the authors offer a method for robust tensor decomposition that can leverage additional rank information to improve recovery accuracy, especially when convex relaxations fall short.

The empirical performance of these models is rigorously evaluated on both synthetic and real-world datasets. Strong numerical results are demonstrated, particularly highlighting situations where the proposed algorithms outperform traditional approaches like robust PCA and classical completion methods in handling gross corruptions and missing data. For instance, the Singleton model achieved near-perfect recovery in scenarios with a strategic ratio of observation to corruption. The transition from poor to near-exact recovery with increasing data support exemplifies the efficacy of their approach.

Practical Implications and Future Developments

The robust tensor recovery techniques offered by this paper have significant implications for practical applications. In areas like image processing, where outliers and corruptions are common, these models build a pathway toward more reliable feature extraction and data interpretation. Additionally, the ability to reconstruct incomplete data with robustness to high-dimensional noise stands to revolutionize applications from surveillance systems to medical imaging, where full data capture may be infeasible.

Theoretically, this work extends the understanding of tensor decomposition beyond matrix-based methods, harnessing the full potential of higher-order structures. The combination of convex and nonconvex models sets the stage for future exploration into adaptive strategies that dynamically determine the best-suited model based on the tensor’s intrinsic properties.

While the immediate future could explore automated parameter tuning to reduce the reliance on manual calibration and enhance usability in dynamic environments, further studies could also investigate integrating machine learning paradigms for predictive adjustments of model constraints, thereby advancing the frontier of robust tensor analysis.

In conclusion, this paper makes a significant contribution to the field of computational multilinear algebra, with robust methodologies that push the envelope of current data recovery techniques, both practically and theoretically. These advancements promise to pivot how robust tensor decompositions are approached and applied across multifarious disciplines.