Smooth PARAFAC Decomposition for Tensor Completion (1505.06611v3)

Abstract: In recent years, low-rank based tensor completion, which is a higher-order extension of matrix completion, has received considerable attention. However, the low-rank assumption is not sufficient for the recovery of visual data, such as color and 3D images, where the ratio of missing data is extremely high. In this paper, we consider "smoothness" constraints as well as low-rank approximations, and propose an efficient algorithm for performing tensor completion that is particularly powerful regarding visual data. The proposed method admits significant advantages, owing to the integration of smooth PARAFAC decomposition for incomplete tensors and the efficient selection of models in order to minimize the tensor rank. Thus, our proposed method is termed as "smooth PARAFAC tensor completion (SPC)." In order to impose the smoothness constraints, we employ two strategies, total variation (SPC-TV) and quadratic variation (SPC-QV), and invoke the corresponding algorithms for model learning. Extensive experimental evaluations on both synthetic and real-world visual data illustrate the significant improvements of our method, in terms of both prediction performance and efficiency, compared with many state-of-the-art tensor completion methods.

• The paper introduces the SPC algorithm that leverages smoothness constraints in PARAFAC decomposition to enhance tensor recovery under heavy missing data.
• It employs total and quadratic variation strategies to adaptively impose smoothness on tensor components, eliminating the need for preset rank limits.
• Experimental results demonstrate significant improvements in visual data recovery, outperforming state-of-the-art methods even at extreme missing data ratios.

Overview of "Smooth PARAFAC Decomposition for Tensor Completion"

In the field of tensor completion, standard low-rank assumptions often fall short when applied to visual data domains, such as images and videos, where data loss can be extensive. The paper "Smooth PARAFAC Decomposition for Tensor Completion" by Yokota et al. proposes employing smoothness constraints alongside low-rank approximations for tensor completion, particularly emphasizing visual data recovery. By introducing smoothness, the paper seeks to address the shortcomings of traditional approaches that rely solely on low-rank assumptions. The methodology presented, dubbed "Smooth PARAFAC Tensor Completion" (SPC), incorporates total variation (SPC-TV) and quadratic variation (SPC-QV) strategies to enforce smoothness.

Key Contributions

The proposed SPC algorithm integrates smooth PARAFAC decomposition, enhancing recovery capabilities in scenarios with high missing data ratios. The two main contributions of the paper are:

1. Smoothness Constraints in PARAFAC Decomposition:
   • The authors utilize two variations of smoothness constraints: total variation (TV) and quadratic variation (QV), applied to tensor components rather than on the output tensor itself. This differentiation leads to a robust enhancement in reconstructing tensors with large portions of missing data.
   • The constraints adaptively enforce various levels of smoothness on different tensor components, allowing the model to capture the inherent smooth structures of visual data more effectively.
2. Efficient Rank Estimation Strategy:
   • SPC does not require a predefined upper bound for tensor rank, a common necessity for many state-of-the-art methods. Instead, the algorithm incrementally increases the rank until a desired fit with the available data is achieved. This process eliminates the need for extensive a priori rank estimation, which is often inaccurate in high data loss scenarios.

Experimental Evaluation

The authors conduct extensive experiments using both synthetic and real-world datasets. The SPC methodology exhibits significant improvements over state-of-the-art methods such as LTVNN, HaLRTC, STDC, and FBCP-MP in terms of prediction performance and processing efficiency.

• Image Completion: The SPC-QV model consistently outperformed traditional methods, especially under extreme missing data ratios (up to 99%). It demonstrated superior performance accuracies by maintaining high visual fidelity when reconstructing images with varying complexities and obstruction levels.
• Real-world Data Applications: Applied to MRI and facial image datasets, SPC showed robustness in handling not only standard missing data scenarios but also in highly irregular ones, where existing methods struggled to maintain data integrity and structural similarity.

Theoretical and Practical Implications

The integration of smoothness constraints in tensor decomposition transcends traditional approaches, which heavily rely on mere low-rank assumptions. The successful application of total and quadratic variations points to potential new exploration avenues in optimizing visual data processing, where preserving intrinsic visual continuity is paramount.

The rank-increasing strategy introduced by SPC challenges conventional rank estimation methods in tensor completion, proposing a model that adapts more naturally to the data’s inherent structure.

Future Directions

The advancement achieved through SPC highlights several future directions for research, particularly in broadening the scope of smoothness applications across other domains, such as bioinformatics or network analysis, where similar data structuring issues are prevalent. Moreover, the paper suggests refining optimization processes in tensor factorization incorporating manifold-based methods for improved computational efficiency and convergence reliability.

In conclusion, Yokota et al.'s work significantly contributes to tensor completion research, presenting a methodology capable of overcoming limitations associated with high-level data losses in visual contexts. The smoothness-enhanced PARAFAC decomposition offers practical insights and avenues for further innovation in data recovery and tensor analysis.