Exact Tensor Completion Using t-SVD

Abstract

Abstract-In this paper we focus on the problem of completion of multidimensional arrays (also referred to as tensors) from limited sampling. Our approach is based on a recently proposed tensor-Singular Value Decomposition (t-SVD) [1]. Using this factorization one can derive notion of tensor rank, referred to as the tensor tubal rank, which has optimality properties similar to that of matrix rank derived from SVD. As shown in [2] some multidimensional data, such as panning video sequences exhibit low tensor tubal rank and we look at the problem of completing such data under random sampling of the data cube. We show that by solving a convex optimization problem, which minimizes the tensor nuclear norm obtained as the convex relaxation of tensor tubal rank, one can guarantee recovery with overwhelming probability as long as samples in proportion to the degrees of freedom in t-SVD are observed. In this sense our results are order-wise optimal. The conditions under which this result holds are very similar to the incoherency conditions for the matrix completion, albeit we define incoherency under the algebraic set-up of t-SVD. We show the performance of the algorithm on some real data sets and compare it with other existing approaches based on tensor flattening and Tucker decomposition.

Low-Rank Tensor Completion by Truncated Nuclear Norm Regularization

“…This suggests that our tensor nuclear norm can be efficiently computed by one matrix SVD in the Fourier domain, instead of using the sophisticated t-SVD to obtain S. Our definition is different from those of previous work [9], [14], which are also calculated in the Fourier domain. The tubal nuclear norm in [9] comprises computing each frontal slice ofS.…”

“…This suggests that our tensor nuclear norm can be efficiently computed by one matrix SVD in the Fourier domain, instead of using the sophisticated t-SVD to obtain S. Our definition is different from those of previous work [9], [14], which are also calculated in the Fourier domain. The tubal nuclear norm in [9] comprises computing each frontal slice ofS. Similarly, Lu et al [14] further proved that the tubal nuclear norm can be computed by the nuclear norm of the block circulant matrix of a tensor with factor 1/n 3 , i.e., ||A|| * = 1 n3 ||bcirc(A)|| * .…”

“…But no theoretical analysis interprets that the nuclear norm of each matricization of X is plausible because the spacial structure may lose by the matricization. Moreover, it is unclear to choose the optimal values of α i [9], though they control the weights of k norms in problem (1). Generally, α i 's are determined empirically.…”

Review of Polynomial Chaos-Based Methods for Uncertainty Quantification in Modern Integrated Circuits

“…Indeed, the application of such techniques has grown rapidly: micromagnetics [96][97][98], model order reduction [99,100], big data [101][102][103], signal processing [104][105][106][107], control design [108,109], and electronic design automation [93].…”

Tensor completion via optimization on the product of matrix manifolds

“…We note that the for this data tubal sampling is worse compared to random sampling but has reasonable performance as the sampling rate increases. While it may be intuitively obvious that random sampling should be better compared to tubal-sampling, the result of tensor completion on panning video can give comparable performance for the same sampling rates, see [17] for such an example (we omit the manifold results for panning video here due to lack of space).…”