Multidimensional Data Recovery Based On Tensor Ring Factor Low-rankness

Low rank tensor completion(LRTC)has attracted extensive attention for its application in computer vision,knowledge graph Completion,power data recovery,etc.The LRTC problem aims to predict or recover missing data from incomplete data,where the key lies in exploring the low-rank structure of the original data.Traditional tensor completion methods tend to consider the low-rankness of tensor unfolded matrix,which destroys the multi-way structure of the original tensor and leads to performance degradation.Therefore,how to find a more compact and efficient low-rank representation from original high-dimensional data is an important research topic in the field of tensor completion.In recent years,tensor ring(TR)decomposition has also received much attention due to its excellent representation of high-order data,which can compress high-order data at a higher compression ratio while maintaining the quality of tensor data.In solving LRTC problems using tensor ring decomposition,the degree of mining the low-rankness of the tensor ring factors often directly affects the final results,and there is little research in this area.Therefore,this thesis investigates the low-rankness of its core factors based on tensor ring decomposition,constructs a more compact decomposition model for the original data,and applies it to multidimensional data recovery.The specific content is as follows:1.Since large non-zero singular values significantly affect the nuclear norm but do not really affect the critical rank,this thesis defines a tensor ring truncation nuclear norm to explore the low-rankness of the tensor ring factor by minimizing the tensor ring nuclear norm.Larger singular values tend to play a dominant role and have little significance for capturing the low-rankness of the tensor ring core factor,so the truncated nuclear norm can attenuate the effect of large singular values and find a better low-rank approximation.In addition,this thesis applies the tensor ring truncation nuclear norm to multidimensional data recovery,provides a theorem to solve the optimization problem of tensor ring truncation nuclear norm and designs a tensor completion optimization algorithm framework based on alternating direction method of multipliers(ADMM).2.For high-order multidimensional data,this thesis develops a hierarchical factorization strategy to preserve the low-rank structure of the original data with a lower number of parameters.In the first layer,the original data is represented by a compact tensor ring with low TR rank;In the second layer,the low rankness of each tensor ring core factor is used to find a compact representation of the tensor ring core factor,so as to capture the more compact low-rank representation of the original data.In addition,this thesis also applies the above strategies to tensor completion problems and proposes an optimization algorithm based on block coordinate descent(BCD).Through this hierarchical factorization model,the low-rank property of each mode of the original tensor can be fully utilized,so as to obtain better tensor completion performance.The convergence of the above algorithms based on ADMM and BCD has corresponding theoretical guarantee,and the subsequent experiments also prove that both proposed algorithms can converge to a stationary point.In addition,a large number of experiments on the recovery of simulated data,images and video data show that the proposed method achieves higher recovery accuracy than similar methods and is more efficient than most algorithms.