Double Transformed Tubal Nuclear Norm Minimization for Tensor Completion

Non-convex methods play a critical role in low-rank tensor completion for their approximation to tensor rank is tighter than that of convex methods.But they usually cost much more time for calculating singular values of large tensors.In this paper,we propose a double transformed tubal nuclear norm(DTTNN)to replace the rank norm penalty in low rank tensor completion(LRTC)tasks.DTTNN turns the original non-convex penalty of a large tensor into two convex penalties of much smaller tensors,and it is shown to be an equivalent transformation.Therefore,DTTNN could take advantage of non-convex envelopes while saving time.Experimental results on color image and video inpainting tasks verify the effectiveness of DTTNN compared with state-of-the-art methods.

Improving POI Recommendation via Non-Convex Regularized Tensor Completion

The problem of low accuracy of POI(Points of Interest)recommendation in LBSN(Location-Based Social Networks)has not been effectively solved.In this paper,a POI recommendation algorithm based on non-convex regularized tensor completion is proposed.The fourth-order tensor is constructed by using the current location category,the next location category,time and season,the regularizer is added to the objective function of tensor completion to prevent over-fitting and reduce the error of the model.The proximal algorithm is used to solve the objective function,and the adaptive momentum is introduced to improve the efficiency of the solution.The experimental results show that the algorithm can improve recommendation accuracy while reducing the time cost.

An Accelerated Proximal Gradient Algorithm for Hankel Tensor Completion

In this paper,an accelerated proximal gradient algorithm is proposed for Hankel tensor completion problems.In our method,the iterative completion tensors generated by the new algorithm keep Hankel structure based on projection on the Hankel tensor set.Moreover,due to the special properties of Hankel structure,using the fast singular value thresholding operator of the mode-s unfolding of a Hankel tensor can decrease the computational cost.Meanwhile,the convergence of the new algorithm is discussed under some reasonable conditions.Finally,the numerical experiments show the effectiveness of the proposed algorithm.

On the robustness of EEG tensor completion methods

During the acquisition of electroencephalographic(EEG) signals, data may be missing or corrupted by noise and artifacts. To reconstruct the incomplete data, EEG signals are firstly converted into a three-order tensor(multi-dimensional data) of shape time × channel × trial. Then, the missing data can be efficiently recovered by applying a tensor completion method(TCM).However, there is not a unique way to organize channels and trials in a tensor, and different numbers of channels are available depending on the EEG setting used, which may affect the quality of the tensor completion results. The main goal of this paper is to evaluate the robustness of EEG completion methods with several designed parameters such as the ordering of channels and trials, the number of channels, and the amount of missing data. In this work, the results of completing missing data by several TCMs were compared. To emulate different scenarios of missing data, three different patterns of missing data were designed.Firstly, the amount of missing data on completion effects was analyzed, including the time lengths of missing data and the number of channels or trials affected by missing data. Secondly, the numerical stability of the completion methods was analyzed by shuffling the indices along channels or trials in the EEG data tensor. Finally, the way that the number of electrodes of EEG tensors influences completion effects was assessed by changing the number of channels. Among all the applied TCMs, the simultaneous tensor decomposition and completion(STDC) method achieves the best performance in providing stable results when the amount of missing data or the electrode number of EEG tensors is changed. In other words, STDC proves to be an excellent choice of TCM, since permutations of trials or channels have almost no influence on the complete results. The STDC method can efficiently complete the missing EEG signals. The designed simulations can be regarded as a procedure to validate whether or not a completion method is useful enough to complete EEG signals.

A hybrid method to select morphometric features using tensor completion and F-score rank for gifted children identification

Gifted children are able to learn in a more advanced way than others, probably due to neurophysiological differences in the communication efficiency in neural pathways. Topological features contribute to understanding the correlation between the brain structure and intelligence. Despite decades of neuroscience research using MRI, methods based on brain region connectivity patterns are limited by MRI artifacts, which therefore leads to revisiting MRI morphometric features, with the aim of using them to directly identify gifted children instead of using brain connectivity. However, the small, high-dimensional morphometric feature dataset with outliers makes the task of finding good classification models challenging. To this end, a hybrid method is proposed that combines tensor completion and feature selection methods to handle outliers and then select the discriminative features. The proposed method can achieve a classification accuracy of 93.1%, higher than other existing algorithms, which is thus suitable for the small MRI datasets with outliers in supervised classification scenarios.

Accurate and Computational Efficient Joint Multiple Kronecker Pursuit for Tensor Data Recovery

This paper addresses the problem of tensor completion from limited samplings.Generally speaking,in order to achieve good recovery result,many tensor completion methods employ alternative optimization or minimization with SVD operations,leading to a high computational complexity.In this paper,we aim to propose algorithms with high recovery accuracy and moderate computational complexity.It is shown that the data to be recovered contains structure of Kronecker Tensor decomposition under multiple patterns,and therefore the tensor completion problem becomes a Kronecker rank optimization one,which can be further relaxed into tensor Frobenius-norm minimization with a constraint of a maximum number of rank-1 basis or tensors.Then the idea of orthogonal matching pursuit is employed to avoid the burdensome SVD operations.Based on these,two methods,namely iterative rank-1 tensor pursuit and joint rank-1 tensor pursuit are proposed.Their economic variants are also included to further reduce the computational and storage complexity,making them effective for large-scale data tensor recovery.To verify the proposed algorithms,both synthesis data and real world data,including SAR data and video data completion,are used.Comparing to the single pattern case,when multiple patterns are used,more stable performance can be achieved with higher complexity by the proposed methods.Furthermore,both results from synthesis and real world data shows the advantage of the proposed methods in term of recovery accuracy and/or computational complexity over the state-of-the-art methods.To conclude,the proposed tensor completion methods are suitable for large scale data completion with high recovery accuracy and moderate computational complexity.

Robust tensor decomposition via orientation invariant tubal nuclear norms

Aiming at recovering an unknown tensor(i.e.,multi-way array)corrupted by both sparse outliers and dense noises,robust tensor decomposition(RTD)serves as a powerful pre-processing tool for subsequent tasks like classification and target detection in many computer vision and machine learning applications.Recently,tubal nuclear norm(TNN)based optimization is proposed with superior performance as compared with other tensorial nuclear norms for tensor recovery.However,one major limitation is its orientation sensitivity due to low-rankness strictly defined along tubal orientation and it cannot simultaneously model spectral low-rankness in multiple orientations.To this end,we introduce two new tensor norms called OITNN-O and OITNN-L to exploit multi-orientational spectral low-rankness for an arbitrary K-way(K≥3)tensors.We further formulate two RTD models via the proposed norms and develop two algorithms as the solutions.Theoretically,we establish non-asymptotic error bounds which can predict the scaling behavior of the estimation error.Experiments on real-world datasets demonstrate the superiority and effectiveness of the proposed norms.

Completed Tensor Product and Flatness

We give,in a more general case than the Noetherian case,an answer to the question posed by Shaul on the flatness of the completed tensor product.

Completely positive tensors in the complex field

In this paper, we introduce the complex completely positive tensor, which has a symmetric complex decomposition with all real and imaginary parts of the decomposition vectors being non-negative. Some properties of the complex completely positive tensor are given. A semidefinite algorithm is also proposed for checking whether a complex tensor is complex completely positive or not. If a tensor is not complex completely positive, a certificate for it can be obtained;if it is complex completely positive, a complex completely positive decomposition can be obtained.

FAST ALGORITHMS FOR HIGHER-ORDER SINGULAR VALUE DECOMPOSITION FROM INCOMPLETE DATA

Higher-order singular value decomposition （HOSVD） is an efficient way for data reduction and also eliciting intrinsic structure of multi-dimensional array data. It has been used in many applications, and some of them involve incomplete data. To obtain HOSVD of the data with missing values, one can first impute the missing entries through a certain tensor completion method and then perform HOSVD to the reconstructed data. However, the two-step procedure can be inefficient and does not make reliable decomposition. In this paper, we formulate an incomplete HOSVD problem and combine the two steps into solving a single optimization problem, which simultaneously achieves imputation of missing values and also tensor decomposition. We also present one algorithm for solving the problem based on block coordinate update （BCU）. Global convergence of the algorithm is shown under mild assumptions and implies that of the popular higher-order orthogonality iteration （HOOI） method, and thus we, for the first time, give global convergence of HOOI. In addition, we compare the proposed method to state-of-the-art ones for solving incom- plete HOSVD and also low-rank tensor completion problems and demonstrate the superior performance of our method over other compared ones. Furthermore, we apply it to face recognition and MRI image reconstruction to show its practical performance.