Efficient tensor completion for color image and video recovery: Low-rank tensor train (1606.01500v1)

Abstract: This paper proposes a novel approach to tensor completion, which recovers missing entries of data represented by tensors. The approach is based on the tensor train (TT) rank, which is able to capture hidden information from tensors thanks to its definition from a well-balanced matricization scheme. Accordingly, new optimization formulations for tensor completion are proposed as well as two new algorithms for their solution. The first one called simple low-rank tensor completion via tensor train (SiLRTC-TT) is intimately related to minimizing a nuclear norm based on TT rank. The second one is from a multilinear matrix factorization model to approximate the TT rank of a tensor, and is called tensor completion by parallel matrix factorization via tensor train (TMac-TT). A tensor augmentation scheme of transforming a low-order tensor to higher-orders is also proposed to enhance the effectiveness of SiLRTC-TT and TMac-TT. Simulation results for color image and video recovery show the clear advantage of our method over all other methods.

• The paper demonstrates that TT rank optimization captures global correlations for accurate tensor completion in color image and video recovery.
• It introduces novel algorithms, SiLRTC-TT and TMac-TT, which outperform traditional CP and Tucker methods even with high data missing rates.
• The study employs a ket augmentation scheme to convert lower-order tensors into higher orders, enabling robust exploitation of latent data structures.

Efficient Tensor Completion for Color Image and Video Recovery: Low-Rank Tensor Train

The paper presents an alternative approach to the tensor completion problem, specifically targeting scenarios involving color image and video recovery. The focus is on leveraging the low-rank tensor train (TT) decomposition to achieve effective tensor completion, offering a departure from more traditional methods that typically utilize the CANDECOMP/PARAFAC (CP) and Tucker decompositions. The proposed methodology showcases several optimization formulations for tensor completion, introducing two novel algorithms, SiLRTC-TT and TMac-TT.

The underlying challenge addressed in this paper revolves around the estimation of missing entries in tensors representing multidimensional datasets—such as color images and videos—by considering the inherent low-rank structure of these data formations. The TT decomposition is put forth as an apt tool for this application due to its ability to better capture global correlations within the data. This stems from its adoption of balanced matricization schemes, which contrasts with the unbalanced approaches commonly associated with Tucker-based completions.

Contributions and Methodology

1. TT Rank Optimization: The authors argue that the TT rank can effectively capture global correlations, making it an appropriate metric for low-rank tensor completion (LRTC). Unlike the Tucker rank, which can often lead to inefficient optimization, TT rank-based optimization offers tractability.
2. Algorithm Development: Two new algorithms, SiLRTC-TT and TMac-TT, were developed to solve the TT rank optimization problem efficiently:
   • SiLRTC-TT applies a TT nuclear norm minimization strategy.
   • TMac-TT uses multilinear matrix factorization, avoiding the direct use of singular value decompositions (SVD), thus enhancing computational efficiency.
3. Ket Augmentation Scheme: This novel technique allows the transformation of lower-order tensors into higher-order representations without altering the total number of entries, thereby enabling a rich and effective exploration of the data's latent structure through TT-based optimizations.

Results

The simulation results on color image and video datasets demonstrated that the proposed methods—especially TMac-TT—outperformed existing methods concerning completion accuracy, particularly when dealing with high levels of missing data. The experiments showcased TMac-TT's robustness, achieving successful completion rates even under conditions wherein 95% of the data entries were missing. This indicates a significant advantage in using TT-based approaches, especially for large-scale and complex datasets typical in image and video applications.

Implications and Future Directions

The research introduces a paradigm in tensor completion by shifting the focus towards TT rank-based methodologies. Practically, this advancement holds potential for substantial improvements in diverse applications, particularly within fields requiring large-scale tensor data recovery such as computer vision, environmental monitoring, and biomedical imaging.

From a theoretical perspective, the paper emphasizes the need to further explore TT decomposition strategies, potentially broadening their applicability beyond traditional mathematical and physical problem domains into mainstream machine learning and data processing tasks.

Future work could explore automatic and adaptive determination of appropriate TT ranks and weight parameters, enhancing the usability of the proposed algorithms in dynamic and real-time applications. Additionally, extending the benefits seen in color image and video recovery to other forms of data—such as hyperspectral images or multi-modal datasets—remains an intriguing avenue for further exploration.

In conclusion, this paper provides a compelling argument for the adoption of TT rank-based tensor completion, offering a viable path towards more effective, efficient, and scalable data recovery solutions in multi-dimensional spaces.