Sparse Nonnegative Tensor Factorization and Completion With Noisy Observations
In this paper, we study the sparse nonnegative tensor factorization and completion problem from partial and noisy observations for third-order tensors. Because of sparsity and nonnegativity, the underlying tensor is decomposed into the tensor-tensor product of one sparse nonnegative tensor and one n...
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Published in | IEEE transactions on information theory Vol. 68; no. 4; pp. 2551 - 2572 |
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Main Authors | , |
Format | Journal Article |
Language | English |
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New York
IEEE
01.04.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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Abstract | In this paper, we study the sparse nonnegative tensor factorization and completion problem from partial and noisy observations for third-order tensors. Because of sparsity and nonnegativity, the underlying tensor is decomposed into the tensor-tensor product of one sparse nonnegative tensor and one nonnegative tensor. We propose to minimize the sum of the maximum likelihood estimation for the observations with nonnegativity constraints and the tensor <inline-formula> <tex-math notation="LaTeX">\ell _{0} </tex-math></inline-formula> norm for the sparse factor. We show that the error bounds of the estimator of the proposed model can be established under general noise observations. The detailed error bounds under specific noise distributions including additive Gaussian noise, additive Laplace noise, and Poisson observations can be derived. Moreover, the minimax lower bounds are shown to be matched with the established upper bounds up to a logarithmic factor of the sizes of the underlying tensor. These theoretical results for tensors are better than those obtained for matrices, and this illustrates the advantage of the use of nonnegative sparse tensor models for completion and denoising. Numerical experiments are provided to validate the superiority of the proposed tensor-based method compared with the matrix-based approach. |
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AbstractList | In this paper, we study the sparse nonnegative tensor factorization and completion problem from partial and noisy observations for third-order tensors. Because of sparsity and nonnegativity, the underlying tensor is decomposed into the tensor-tensor product of one sparse nonnegative tensor and one nonnegative tensor. We propose to minimize the sum of the maximum likelihood estimation for the observations with nonnegativity constraints and the tensor [Formula Omitted] norm for the sparse factor. We show that the error bounds of the estimator of the proposed model can be established under general noise observations. The detailed error bounds under specific noise distributions including additive Gaussian noise, additive Laplace noise, and Poisson observations can be derived. Moreover, the minimax lower bounds are shown to be matched with the established upper bounds up to a logarithmic factor of the sizes of the underlying tensor. These theoretical results for tensors are better than those obtained for matrices, and this illustrates the advantage of the use of nonnegative sparse tensor models for completion and denoising. Numerical experiments are provided to validate the superiority of the proposed tensor-based method compared with the matrix-based approach. In this paper, we study the sparse nonnegative tensor factorization and completion problem from partial and noisy observations for third-order tensors. Because of sparsity and nonnegativity, the underlying tensor is decomposed into the tensor-tensor product of one sparse nonnegative tensor and one nonnegative tensor. We propose to minimize the sum of the maximum likelihood estimation for the observations with nonnegativity constraints and the tensor <inline-formula> <tex-math notation="LaTeX">\ell _{0} </tex-math></inline-formula> norm for the sparse factor. We show that the error bounds of the estimator of the proposed model can be established under general noise observations. The detailed error bounds under specific noise distributions including additive Gaussian noise, additive Laplace noise, and Poisson observations can be derived. Moreover, the minimax lower bounds are shown to be matched with the established upper bounds up to a logarithmic factor of the sizes of the underlying tensor. These theoretical results for tensors are better than those obtained for matrices, and this illustrates the advantage of the use of nonnegative sparse tensor models for completion and denoising. Numerical experiments are provided to validate the superiority of the proposed tensor-based method compared with the matrix-based approach. |
Author | Ng, Michael K. Zhang, Xiongjun |
Author_xml | – sequence: 1 givenname: Xiongjun orcidid: 0000-0002-5387-4129 surname: Zhang fullname: Zhang, Xiongjun email: xjzhang@mail.ccnu.edu.cn organization: School of Mathematics and Statistics and the Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, China – sequence: 2 givenname: Michael K. orcidid: 0000-0001-6833-5227 surname: Ng fullname: Ng, Michael K. email: mng@maths.hku.hk organization: Department of Mathematics, The University of Hong Kong, Hong Kong |
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SubjectTerms | Additives error bound Factorization Gaussian noise Lower bounds Mathematical analysis Maximum likelihood estimation Minimax technique Noise measurement Numerical models Random noise Sparse matrices Sparse nonnegative tensor factorization and completion tensor-tensor product Tensors Upper bound Upper bounds |
Title | Sparse Nonnegative Tensor Factorization and Completion With Noisy Observations |
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