High-order Data Completion Via Low-rank Tensor Ring:Models And Algorithms

With the rapid development of society,high-order data(e.g.hyperspectral images,color images,videos)frequently appears in many modern scientific and applied fields.However,due to the unpredictable reasons such as mechanical failure of sampling equipment and improper data storage,data is often faced with the problem of missing data in every link from collection to the final presentation.And the lack of data will directly affect the performance of data analysis in practical applications.Therefore,how to effectively recover the missing data has become one of the important research hotspots.Due to the natural advantages of tensors in preserving the multimodal structure of higher-order data,many low-rank tensor learning methods are widely used to solve the higher-order data completion problem that is also known as the tensor completion problem in the field of linear algebra.Compared with the classical low-rank tensor learning methods,such as CAN-DECOMP/PARAFAC(CP),Tucker,Tensor-Train(TT)decompositions,the recently proposed method called Tensor-Ring decomposition was shown to be more powerful in the low-rank characterization of high-order data,which has become an important research direction in tensor completion.Therefore,based on Tensor-Ring decomposition,this paper studies the low-rank theory of tensors,captures the low-rank structure of high-order missing data,establishes ten-sor completion models,and designs effective optimization algorithms.Spcifically,the research results of this paper are as follows:Firstly,to reduce the highly computational cost of tensor-ring complection,we propose an efficient tensor-ring completion model based on parallel matrix factorization.Due to an un-balanced unfolding scheme used during the update of core tensors,the conventional TR-based completion methods usually require a large TR-rank to achieve the optimal performance,which leads to highly computational cost in practicall applications.To overcome this drawback,we first define a balanced unfolding operation called tensor circular unfolding(TCU),by which the relationship between TR-rank and the ranks of tensor unfoldings is theoretically established.Using this new unfolding operation,we further propose an algorithm to exploit the low TR-rank structure by performing parallel low-rank matrix factorizations to all circularly-unfolded matri-ces.To tackle the problem of non-uniform missing patterns,we apply a row weighting method to each circularly-unfolded matrix,which significantly improves the adaptive ability to vari-ous types of missing patterns.Compared with conventional TR-based completion algorithms,extensive experiments illustrate that the proposed algorithm is able to achieve outstanding per-formance using a much smaller TR-rank,which substantially reduces the computational cost in practical applications.Secondly,to overcome the difficulty of TR-rank selection,we propose a tensor-ring nu-clear norm based completion model.The most existing tensor-ring decomposition based com-pletion models are non-convex,and thus fails to obtain the optimal TR-rank;further,since the TR-rank is defined as a vector,the range of TR-rank selection increases exponentially with TR-rank dimension,which leads to the difficulty to find the optimal TR-rank in practical ap-plications.To overcome this problem,a convex tensor-ring nuclear norm based completion model is proposed.Specifically,by using the tensor circular unfolding operation,we first de-fine a tensor-ring nuclear norm by a weighted combination of tensor circular unfoldings,and then propose a tensor-ring nuclear norm based completion model.Since the tensor-ring nuclear norm is convex and doesn't require predefined rnak,the proposed model avoids the difficulty of TR-rank selection.Extensive experimental results demonstrate that the proposed tensor com-pletion method outperforms the conventional tensor completion methods in the image/video completion.Thirdly,considering the low-rank unbalance of tensor modes,we propose a latent tensor-ring nuclear norm based completion model.The most existing tensor-ring completion methods often assume tha the target tensor has low-rank structure in every mode.However,this strong assumption tends to lead the poor performance for the target tensor with only several low-rank modes.To overcome this drawback,we apply the tensor circular unfolding operation to de-fine the latent tensor-ring nuclear norm,and then develop a Frank-Wolfe based algorithm to optimize it.Since we utilize the sparsity structure of observed tensor in Frank-Wolfe scheme,the proposed method is much lower than other methods in time and space complexities.Ex-tensive experimental results of visual data inpainting demonstrate that the proposed method is effective for the tensor with only several low-rank modes,meanwhile can achieve a rather good performance at smaller costs of time and space.Forthly,to reduce the sensitivity of the tensor-ring completion model to TR-rank selection,we propose a low-rank sparse tensor-ring completion model.Most of the existing TR-based methods tend to suffer from deterioration when the selected rank larger than the true one.To ad-dress this key issue,this thesis proposes a new low-rank sparse Tensor-Ring completion model by imposing the Frobenius norm regularization to latent TR-cores.We theoretically establish that the proposed model is capable of exploiting the low-rank sparse structure using the Frobe-nius norm of latent TR-cores.Moreover,an effective initialization algorithm is proposed and has been demonstrated to speed up the convergence of the proposed model.Compared with other traditional Tensor-Ring based completion methods and other state-of-the-art methods,extensive experimental results on synthetic and real-world data demonstrate the remarkable performance of the proposed method for recovering missing entries and effective robustness when the selected TR-rank increases.