Parallel matrix factorization for low-rank tensor completion (1312.1254v2)

Abstract: Higher-order low-rank tensors naturally arise in many applications including hyperspectral data recovery, video inpainting, seismic data recon- struction, and so on. We propose a new model to recover a low-rank tensor by simultaneously performing low-rank matrix factorizations to the all-mode ma- tricizations of the underlying tensor. An alternating minimization algorithm is applied to solve the model, along with two adaptive rank-adjusting strategies when the exact rank is not known. Phase transition plots reveal that our algorithm can recover a variety of synthetic low-rank tensors from significantly fewer samples than the compared methods, which include a matrix completion method applied to tensor recovery and two state-of-the-art tensor completion methods. Further tests on real- world data show similar advantages. Although our model is non-convex, our algorithm performs consistently throughout the tests and give better results than the compared methods, some of which are based on convex models. In addition, the global convergence of our algorithm can be established in the sense that the gradient of Lagrangian function converges to zero.

• The paper introduces a parallel matrix factorization model that recovers low-rank tensors from incomplete data by simultaneously factorizing all mode unfoldings.
• It employs an alternating minimization algorithm with adaptive rank adjustments to enhance recovery accuracy, outperforming nuclear-norm minimization and single-mode methods.
• Experimental results on synthetic and real-world datasets, including brain MRI and video inpainting, demonstrate lower relative errors and robust performance under sparse sampling and noise.

Parallel Matrix Factorization for Low-Rank Tensor Completion

The paper "Parallel Matrix Factorization for Low-Rank Tensor Completion" by Y. Xu, R. Hao, W. Yin, and Z. Su presents a novel approach to the complex task of low-rank tensor completion (LRTC). LRTC problems emerge in various domains such as hyperspectral data recovery, video inpainting, and seismic data reconstruction. This paper focuses on recovering higher-order tensors from incomplete data by advancing matrix factorization techniques.

The authors propose a model that leverages simultaneous low-rank matrix factorizations of all mode unfoldings of a tensor. To solve the model, an alternating minimization algorithm is applied alongside two adaptive rank-adjusting strategies. These strategies adjust the rank estimates, either decreasing from overestimated ranks or increasing from underestimated ranks. The paper showcases the efficacy of this approach through comprehensive evaluations on both synthetic datasets and real-world scenarios.

Key Contributions and Results

The key contribution of the paper is the introduction of an efficient model for tensor completion that relies on parallel matrix factorizations. The alternating least squares method is employed to solve the proposed model. The paper provides robust evidence that this method outperforms traditional methods by obtaining more accurate tensor reconstructions from fewer samples.

The numerical experiments conducted demonstrate the model's superiority over methods like nuclear-norm minimization and matrix completion methods that use a single mode of tensor matricization, such as FaLRTC, MatComp, and SquareDeal. The results indicate that the phase transition plots underline the method's improved recoverability, capable of handling a wider array of low-rank tensor classes.

Numerical Results:

• Synthetic Data Tests: The proposed model demonstrated significant improvements in phase transition plots over competing methods on 3-way and 4-way tensors with factors having random Gaussian and uniform distributions. The method consistently achieved lower relative errors.
• Real-World Data: For tasks involving real-world data such as brain MRI recoveries and video-inpainting tasks, the model delivered high-quality reconstructions efficiently. Particularly notable is its success at low sample rates and in noisy conditions.

Theoretical Implications and Future Directions

The method's design means it is inherently non-convex, which initially poses challenges in guaranteeing global convergence theoretically. However, the subsequence convergence is established, meaning any limit point satisfies the Karush-Kuhn-Tucker (KKT) conditions. This is an essential metric for ensuring robust performance across varied datasets.

Looking forward, the paper indicates potential enhancements by integrating additional techniques to bolster convergence speed, such as advanced acceleration methods. Additionally, incorporating more diverse structures in the tensor, akin to those hinted by recent developments in structured sparsity, could further enhance outcome quality.

Practical Applications

The practical applications of the research are vast. Efficient and accurate tensor completion methods can significantly impact fields such as medical imaging for faster and more accurate diagnostic analysis, video editing, and data analytics. The model's ability to integrate with large-scale datasets makes it a valuable tool in industries grappling with incomplete or noisy high-dimensional data.

In conclusion, the paper presented by Xu et al. marks a significant step in tensor completion research. By applying parallel matrix factorizations, the method circumvents some existing challenges, offering fresh possibilities for low-rank tensor recovery across various scientific and engineering domains.