| With the advent of the Big Data era,the amount of data to be processed is increas-ing day by day,making data processing and analysis more difficult.High-order tensors can represent large-scale data very well.Tensor decomposition has become one of the effective methods for high-dimensional data compression and processing.The tensors generated in applications are often unbalanced and determining their rank is a NP-hard problem,which makes the tensor decomposition more difficult.The third-order ten-sor is the simplest and most common type of high-order tensors,and the study of the third-order unbalanced tensor decomposition is the basis for studying higher-order ten-sor decomposition.In this thesis,we propose the Automatic Rank-selected Tucker2 decomposition algorithm and the Randomized Automatic Rank-selected Tucker2 de-composition algorithm for third-order unbalanced tensors.The main two parts of the work are as follows:1.The problem of decomposing a third-order unbalanced tensor is transformed into an optimization problem which minimizes the number of elements in the tensor decomposition under the relative error constraint.The solution is constructed based on the alternating optimization,and an automatic rank selection method is proposed.Based on the above methods,we propose the Automatic Rank-selected Tucker2 decomposition(Auto Rank TK2)algorithm.The local optimality and monotonic convergence of the algorithm are proved theoretically.Besides,we test the effectiveness and robustness of the proposed Auto Rank TK2 algorithm via several numerical experiments.2.We apply the randomized SVD to accelerate the Auto Rank TK2 algorithm,and design the corresponding automatic rank selection method.Therefore,the Randomized Automatic Rank-selected Tucker2 decomposition(R-Auto Rank TK2)algorithm is pro-posed.Theoretically,the upper bound of the residual expectation of the R-Auto Rank TK2 algorithm is given.Furthermore,we construct comprehensive numerical experi-ments to verify the acceleration effect of the algorithm. |
