Nonnegative tensor ring(NTR) decomposition is a powerful tool for capturing the significant features of tensor objects while preserving the multi-linear structure of tensor data. The existing algorithms rely on freque...Nonnegative tensor ring(NTR) decomposition is a powerful tool for capturing the significant features of tensor objects while preserving the multi-linear structure of tensor data. The existing algorithms rely on frequent reshaping and permutation operations in the optimization process and use a shrinking step size or projection techniques to ensure core tensor nonnegativity, which leads to a slow convergence rate, especially for large-scale problems. In this paper, we first propose an NTR algorithm based on the modulus method(NTR-MM), which constrains core tensor nonnegativity by modulus transformation. Second, a low-rank approximation(LRA) is introduced to NTR-MM(named LRA-NTR-MM), which not only reduces the computational complexity of NTR-MM significantly but also suppresses the noise. The simulation results demonstrate that the proposed LRA-NTR-MM algorithm achieves higher computational efficiency than the state-of-the-art algorithms while preserving the effectiveness of feature extraction.展开更多
This paper investigates the connections between ring theory, module theory, and graph theory through the graph G(R)of a ring R. We establish that vertices of G(R)correspond to modules, with edges defined by the vanish...This paper investigates the connections between ring theory, module theory, and graph theory through the graph G(R)of a ring R. We establish that vertices of G(R)correspond to modules, with edges defined by the vanishing of their tensor product. Key results include the graph’s connectivity, a diameter of at most 3, and a girth of at most 7 when cycles are present. We show that the set of modules S(R)is empty if and only if R is a field, and that for semisimple rings, the diameter is at most 2. The paper also discusses module isomorphisms over subrings and localization, as well as the inclusion of G(T)within G(R)for a quotient ring T, highlighting that the reverse inclusion is not guaranteed. Finally, we provide an example illustrating that a non-finitely generated module M does not imply M⊗M=0. These findings deepen our understanding of the interplay among rings, modules, and graphs.展开更多
With the advent of tensor-valued time series data,tensor autoregression appears in many fields,in which the coefficient estimation is confronted with the problem of dimensional disaster.Based on the tensor ring(TR)dec...With the advent of tensor-valued time series data,tensor autoregression appears in many fields,in which the coefficient estimation is confronted with the problem of dimensional disaster.Based on the tensor ring(TR)decomposition,an autoregression model with one order for tensor-valued responses is proposed in this paper.A randomized method,TensorSketch,is applied to the TR autoregression model for estimating the coefficient tensor.Convergence and some properties of the proposed methods are given.Finally,some numerical experiment results on synthetic data and real data are given to illustrate the effectiveness of the proposed method.展开更多
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.62073087,61973087 and 61973090)the Key-Area Research and Development Program of Guangdong Province(Grant No.2019B010154002)。
文摘Nonnegative tensor ring(NTR) decomposition is a powerful tool for capturing the significant features of tensor objects while preserving the multi-linear structure of tensor data. The existing algorithms rely on frequent reshaping and permutation operations in the optimization process and use a shrinking step size or projection techniques to ensure core tensor nonnegativity, which leads to a slow convergence rate, especially for large-scale problems. In this paper, we first propose an NTR algorithm based on the modulus method(NTR-MM), which constrains core tensor nonnegativity by modulus transformation. Second, a low-rank approximation(LRA) is introduced to NTR-MM(named LRA-NTR-MM), which not only reduces the computational complexity of NTR-MM significantly but also suppresses the noise. The simulation results demonstrate that the proposed LRA-NTR-MM algorithm achieves higher computational efficiency than the state-of-the-art algorithms while preserving the effectiveness of feature extraction.
文摘This paper investigates the connections between ring theory, module theory, and graph theory through the graph G(R)of a ring R. We establish that vertices of G(R)correspond to modules, with edges defined by the vanishing of their tensor product. Key results include the graph’s connectivity, a diameter of at most 3, and a girth of at most 7 when cycles are present. We show that the set of modules S(R)is empty if and only if R is a field, and that for semisimple rings, the diameter is at most 2. The paper also discusses module isomorphisms over subrings and localization, as well as the inclusion of G(T)within G(R)for a quotient ring T, highlighting that the reverse inclusion is not guaranteed. Finally, we provide an example illustrating that a non-finitely generated module M does not imply M⊗M=0. These findings deepen our understanding of the interplay among rings, modules, and graphs.
基金the handling editor and anonymous referees for useful comments and suggestions which contribute to improving the quality of the manuscriptcrossing research project of Shanghai University of Engineering Science(No.SL-O01).
文摘With the advent of tensor-valued time series data,tensor autoregression appears in many fields,in which the coefficient estimation is confronted with the problem of dimensional disaster.Based on the tensor ring(TR)decomposition,an autoregression model with one order for tensor-valued responses is proposed in this paper.A randomized method,TensorSketch,is applied to the TR autoregression model for estimating the coefficient tensor.Convergence and some properties of the proposed methods are given.Finally,some numerical experiment results on synthetic data and real data are given to illustrate the effectiveness of the proposed method.
