Low-rank tensor recovery using sparse prior and multi-modal tensor factorization
Objective The large amount of data obtained by various terminal devices often results in incomplete data due to missing information or is frequently plagued by degradation issues.Low-rank tensor completion has received significant attention for recovering contaminated data.Tensor decomposition can effectively explore the essential features of tensors,but the tensor rank function induced by traditional tensor decomposition methods cannot explore the correlation between dif-ferent modes of tensors.In addition,traditional tensor completion methods typically impose the total variational constraint on the overall tensor data,which cannot fully utilize the smoothing prior for the low-dimensional subspace of tensors.To address the above two problems,this study proposes a low-rank tensor recovery algorithm using sparse prior and multimode tensor factorization.The traditional low-rank tensor completion models based on tensor rank minimization restore tensors by directly minimizing the tensor rank,in which the tensor rank can be Tucker rank and tensor nuclear norm(TNN).How-ever,extensive research has shown that a correlation exists among different modes of tensor data.The Tucker rank induced by Tucker decomposition and the TNN induced by tensor singular value decomposition cannot flexibly handle multimode correlations within tensors.Therefore,we consider introducing multimode tensor decomposition via mode-n product,incor-porating multimode tensor decomposition into the tensor rank minimization model.In the process of continuous iteration to complete the overall tensor,our model can effectively explore the characteristics of mutual correlation between different modes of the tensor,which can address the limitation of traditional TNN in inadequately capturing the intermode correla-tions within the tensor.Each factor matrix obtained from the multimode tensor decomposition framework encapsulates latent information corresponding to its respective mode,revealing valuable correlated auxiliary information within and across modes,such as the local sparsity exhibited by natural tensor data.By showing that the majority of factor gradients in the factor gradient histogram are zero or close to zero,we can demonstrate that the factors in multimode tensor decomposition exhibit local sparsity.Therefore,on the basis of the assumption of tensor subspace,we consider introducing the local spar-sity prior to preserve the similarity in local segments.Method The method incorporates multimode tensor factorization and local sparsity of decomposed factors based on the tensor rank minimization model.First,the nuclear norm constraint is imposed on the original tensor to capture the global low rankness of the tensor,which makes the model robust when dealing with tensor completion tasks.Second,multimode tensor factorization is used to decompose the tensor into a series of low-dimensional tensors and a series of factor matrices along each mode,which explores the correlation between different modes.The factor gradient sparsity regularization constraint is imposed on the factor matrices to explore the local sparsity of the tensor subspace,which further improves the tensor recovery performance.Specifically,after tensor decomposition,first-order differencing is applied,and the norm smoothness constraint is leveraged.Combining multimode tensor decompo-sition with tensor subspace sparsity,a robust tensor completion model is developed.The proposed model is optimized through the alternating direction method of multipliers(ADMM)framework,which is achieved by iteratively updating vari-ous variables to accomplish tensor completion and tensor decomposition simultaneously.Result The method in this paper is quantitatively and qualitatively compared with eight other restoration methods at three loss rates on hyperspectral images,multispectral images,YUV(also known as YCbCr)videos,and medical imaging data.The restoration effect of our method is basically the same as that of the deep learning GP-WLRR method,but it has no computational burden at all.Compared with six other tensor modeling methods,our method achieves the best results in terms of mean peak signal-to-noise ratio(MPSNR)and mean structural similarity(MSSIM)metrics.It exhibits superior recovery performance even at high loss rates up to 95％.This finding demonstrates the effectiveness of the proposed model in tensor data recovery.Conclusion The low-rank ten-sor completion algorithm based on sparse prior and multimode tensor decomposition proposed in this paper can simultaneously exploit the global low rankness and local sparsity of a tensor and effectively recover contaminated multichannel visual data.
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