- Tensor N-tubal rank and its convex relaxation for low-rank tensor recovery. - Low-rank tensor recovery (LRTR) Mode-k 1 k 2 tensor unfolding Tensor N-tubal rank. - To tackle these two issues, we define a new tensor unfolding operator, named mode-k 1 k 2 tensor unfolding, as the process of lexicographically stacking all mode-k 1 k 2 slices of an N-way tensor into a three-way tensor, which is a three-way exten- sion of the well-known mode-k tensor matricization. - On this basis, we define a novel ten- sor rank, named the tensor N-tubal rank, as a vector consisting of the tubal ranks of all mode-k 1 k 2 unfolding tensors, to depict the correlations along different modes. - To effi- ciently minimize the proposed N-tubal rank, we establish its convex relaxation: the weighted sum of the tensor nuclear norm (WSTNN). - The key to tensor recovery is to explore the redundancy prior of the underlying tensor, which is usually formulated as low-rankness. - A conclusive issue of LRTR is the definition of the tensor rank. - However, unlike the matrix rank, the definition of the tensor rank is not unique. - The CP rank and the Tucker rank are the two most typical definitions of the tensor rank. - Although the measure of the CP rank is consistent with that of the matrix rank, it is difficult to establish a solvable relaxation form. - The Tucker rank is defined as a vector, the k-th element of which is the rank of the mode-k unfolding matrix [20], i.e.,. - Thus, the SNN faces dif- ficulty in preserving the intrinsic structure of the tensor. - As a generalization of the matrix singular value decomposition (SVD), t-SVD regards a three- way tensor X as a matrix, each element of which is a tube (mode-3 fiber), and then decomposes X as. - Specifically, the tensor tubal rank is equal to the maximum value of the tensor multi-rank. - However, most real-world data always have differ- ent correlations along different modes, e.g., the correlation of an HSI along its spectral mode should be much stronger than those along its spatial modes. - To apply t-SVD to N-way tensors (N P 3), in this paper, we define a three-way extension of the tensor matricization oper- ator, named mode-k 1 k 2 tensor unfolding (k 1 <. - To characterize the correlations along different modes in a more flexible manner, we propose a new tensor rank, named the tensor N-tubal rank, which is a vector consisting of the tubal ranks of all mode-k 1 k 2 unfolding tensors, i.e.,. - Table 1 compares the Tucker rank and the N-tubal rank of two HSIs. - To efficiently minimize the proposed tensor N-tubal rank, we establish its convex relaxation: the weighted sum of the tensor nuclear norm (WSTNN), which can be expressed as the weighted sum of the TNN of each mode-k 1 k 2 unfolding tensor, i.e.,. - Numerous numer- ical experiments on synthetic and real-world data are conducted to illustrate the effectiveness and efficiency of the proposed methods.. - Section 3 gives the defi- nitions of the tensor N-tubal rank and its convex surrogate WSTNN. - Section 5 evaluates the performance of the proposed models and compares the results with those of state-of-the-art competing methods. - 1 The rank is approximated by the numbers of singular values larger than 1% of the largest ones.. - The conjugate transpose of a three-way tensor X 2 R n 1 n 2 n 3 , denoted as X T , is the tensor obtained by conjugate transposing each of the frontal slices and then reversing the order of transposed frontal slices 2 through n 3 . - Data Size Tucker rank N-tubal rank. - Now, we give the definitions of the tensor multi-rank and tubal rank.. - The tensor multi-rank of X is a vector rank m ð Þ 2 X R n 3 , the i-th element of which is the rank of the i-th frontal slice of X, where X ¼ fft ð X . - The tensor nuclear norm of a tensor X 2 R n 1 n 2 n 3 , denoted as kXk TNN , is defined as the sum of the singular values of all the frontal slices of X, i.e.,. - Tensor N -tubal rank and convex relaxation. - In this section, we first propose the mode-k 1 k 2 tensor unfolding operation and then give the definitions of the tensor N- tubal rank and its convex relaxation WSTNN.. - Illustration of the t-SVD of an n 1 n 2 n 3 tensor.. - the frontal slices of which are the lexicographic orderings of the mode-k 1 k 2 slices of X.. - 2 R N N1 ð Þ=2 : Clearly, for a three-way tensor, the tensor tubal rank is the first element of the tensor N-tubal rank. - Specifically, the proposed N-tubal rank combines the advantages of the Tucker rank and tubal rank. - On the one hand, compared with the mode-k 1 unfolding matrix, the mode-k 1 k 2 unfolding tensor avoids the destruction of the structure infor- mation along the k 2 -th mode. - 2, the tubal rank of each mode-k 1 k 2 unfolding (permuta- tion) tensor X ð k 1 k 2 Þ more directly depicts the correlation of the k 1 -th and the k 2 -th modes, i.e., it lacks direct characterization of the correlation along other modes. - The following theorem reveals the relationship between the tensor N-tubal rank and Tucker rank.. - Illustration of the low N-tubal rank prior of an HSI. - (e) Singular value curves of the first frontal slices of X ð k 1 k 2 Þ. - Then, each element of the N-tubal rank is bounded by the Tucker rank along the corresponding modes, i.e.,. - tubal Rank X ð k 1 k 2 Þ. - Then, the N-tubal rank of X is at most r ones N N ð ð 1 Þ = 2 . - The WSTNN of an N-way tensor X 2 R n 1 n 2 n N , denoted as kXk WSTNN , is defined as the weighted sum of the TNN of each mode-k 1 k 2 unfolding tensor, i.e.,. - For the choice of the weight a , we consider the following three cases.. - Case 1: The tensor N-tubal rank of the underlying tensor is unknown and cannot be estimated empirically, such as the case of MRI data. - Case 2: The tensor N-tubal rank of the underlying tensor X 2 R n 1 n 2 n N is known, i.e., N rank t ð Þ ¼ X ð r 11 . - Since a k 1 k 2 stands for the contribution of the TNN of the mode-k 1 k 2 unfolding tensor X ð k 1 k 2 Þ , the value of a k 1 k 2 should be dependent on the tubal rank of X ð k 1 k 2 Þ (r k 1 k 2 ) and the size of the first two modes of X ð k 1 k 2 Þ n k 1 andn k 2. - This implies that the value of the first element of the N-tubal rank should be much larger than the values of its second and third elements. - Considering an N- way tensor X 2 R n 1 n 2 n N , the proposed WSTNN-based LRTC model is formulated as. - Algorithm 2 ADMM-based optimization algorithm for the proposed WSTNN-based LRTC model (12).. - Within the framework of the ADMM, Y k 1 k 2 . - The pseudocode of the developed algorithm is described in Algorithm 2.. - We analyse the computational complexity of the developed algorithm, which involves three subproblems, i.e., the Y k 1 k 2. - Considering an N-way tensor X 2 R n 1 n 2 n N , the proposed WSTNN-based TRPCA model can be formulated as. - Algorithm 3 ADMM-based optimization algorithm for the proposed WSTNN-based TRPCA model (22).. - The pseudocode of the proposed algorithm for solving the proposed WSTNN-based TRPCA model (22) is described in Algorithm 3.. - We analyse the detailed computational complexity of the developed algorithm, which involves five subproblems, i.e., the Z k 1 k 2 subproblems, the L subproblem, the E subproblem, the P k 1 k 2 subproblem, and the M subproblems. - We evaluate the performance of the proposed WSTNN-based LRTC and TRPCA methods. - We employ the peak signal-to-noise rate (PSNR), the structural similarity (SSIM) [33], and the feature similarity (FSIM) [41] to measure the quality of the recovered results. - The compared LRTC meth- ods are as follows: HaLRTC [24] and LRTC-TVI [23], representing the state of the art for the Tucker-decomposition-based method. - BCPF [44], representing the state of the art for the CP-decomposition-based method. - and logDet [14], TNN [43], PSTNN [16], and t-TNN [12], representing the state of the art for the t-SVD-based method. - Table 2 shows the parameter settings for the proposed WSTNN-based LRTC method on dif- ferent data.. - The tested synthetic tensors consist of the sum of r rank-one tensors, which are generated by finding the vector outer product on N (N ¼ 3 or 4) random vectors. - In practice, the data in each test are regenerated and confirmed to meet the conditions of Theorem 3, i.e., the N-tubal rank is r ones ð N N ð 1 Þ = 2 . - We define the success rate as the ratio of successful times to the total number of times, where one test is successful if the relative square error of the recovered tensor X ^ and the ground-truth tensor X, i.e., k X Xk ^ 2 F = kXk 2 F , is less than 10 3. - We test data with different N-tubal ranks and sampling rates (SRs), which is defined as the proportion of the known ele- ments. - 3 The codes of the WSTNN-based LRTC and TRPCA methods are available at https://yubangzheng.github.io/.. - Table 3 lists the mean values of the PSNR, SSIM, and FSIM for all 32 MSIs recovered by different LRTC methods. - Table 4 lists the values of the PSNR, SSIM, and FSIM of the tested MRI recovered by the different LRTC methods. - Table 5 lists the values of the PSNR, SSIM, and FSIM of the tested CV recovered by different LRTC methods. - The left two are the results of the TNN-based LRTC method [43] and the proposed WSTNN-based LRTC method on three-way tensors. - The right two are the results of the TNN-based LRTC method [43] and the proposed WSTNN-based LRTC method on four-way tensors. - Parameter settings of the proposed WSTNN-based LRTC method on different data.. - observed, the proposed method has an overall better performance than that of the compared ones with respect to all evaluation indices. - Table 6 lists the values of the PSNR, SSIM, and FSIM of the tested HSV recovered by different LRTC methods. - The completion results of the MRI data with SR ¼ 20%. - In this section, we evaluate the performance of the proposed WSTNN-based TRPCA method by synthetic data and HSI denoising. - Table 8 lists the PSNR, SSIM, and FSIM values of the tested HSIs recovered by different methods. - The completion results at the 49-th frame of the CV news with SR ¼ 10%.. - As observed, our WSTNN-based TRPCA method achieves the best visual results among those of the three compared methods in terms of both noise removal and detail preservation.. - In this section, we discuss the effects of the threshold parameter s and the convergence of the proposed ADMM in the proposed LRTC and TRPCA problems. - Effects of the threshold parameter. - N-tubal rank. - The left two are the results of the TNN-based TRPCA method [25] and the proposed WSTNN-based TRPCA method on three-way tensors. - The right two are the results of the TNN-based TRPCA method [25] and the proposed WSTNN-based TRPCA method on four-way tensors. - Parameter settings of the proposed WSTNN-based TRPCA method on different data.. - 10 Þ), the singular values obtained after performing the t-SVT (in Theorem 4) contain corrupted information, which is not consis- tent with the low-rank prior of the underlying tensor. - Owing to the use of the ADMM framework and the convexity of the objective functions, the con- vergence of the two developed algorithms is guaranteed theoretically. - To illustrate the effectiveness of the proposed N-tubal rank and WSTNN, we applied the WSTNN to two typical LRTR prob- lems, i.e., LRTC and TRPCA problems, and proposed the WSTNN-based LRTC and TRPCA models. - The numerical results demonstrated that the proposed method effectively exploits the correlations along all modes while preserving the intrinsic structure of the underlying tensor.. - One chal- lenging example is the missing slice problem, which usually results in observed data with a lower N-tubal rank than that of the original data. - Second, we plan to establish some nonconvex relaxations to further improve the performance of the proposed method. - The denoising results of the HSIs Washington DC Mall and Pavia University with NL ¼ 0:4. - The completion results of the HSIs Washington DC Mall and Pavia University with SR ¼ 5%. - Thus, the tubal rank of X ð k 1 k 2 Þ (the ð k 1 . - k 2 Þ-th element of the N-tubal rank of X) is at most r, and it is equal to r if the aforementioned conditions are satisfied.. - Table 9 lists the values of the PSNR, SSIM, and FSIM of these two tested HSIs recovered by different LRTC methods. - 12 shows the PSNR, SSIM and FSIM values of each band of the recovered HSI Washington DC Mall obtained by the eight compared LRTC methods with SR ¼ 5. - The PSNR, SSIM, and FSIM values of each band of the recovered HSI Washington DC Mall output by the eight LRTC methods with SR ¼ 5%.
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