Dimension-free Bounds for Covariance Estimation with Tensor-Train Structure
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912892742270976 |
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| author | Patarusau, Artsiom Puchkin, Nikita Rakhuba, Maxim Noskov, Fedor |
| author_facet | Patarusau, Artsiom Puchkin, Nikita Rakhuba, Maxim Noskov, Fedor |
| contents | We consider a problem of covariance estimation from a sample of i.i.d. high-dimensional random vectors. To avoid the curse of dimensionality, we impose an additional assumption on the structure of the covariance matrix $Σ$. To be more precise, we study the case when $Σ$ can be approximated by a sum of double Kronecker products of smaller matrices in a tensor train (TT) format. Our setup naturally extends widely known Kronecker sum and CANDECOMP/PARAFAC models but admits richer interaction across modes. We suggest an iterative polynomial time algorithm based on TT-SVD and higher-order orthogonal iteration (HOOI) adapted to Tucker-2 hybrid structure. We derive non-asymptotic dimension-free bounds on the accuracy of covariance estimation taking into account hidden Kronecker product and tensor train structures. The efficiency of our approach is illustrated with numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08174 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dimension-free Bounds for Covariance Estimation with Tensor-Train Structure Patarusau, Artsiom Puchkin, Nikita Rakhuba, Maxim Noskov, Fedor Statistics Theory We consider a problem of covariance estimation from a sample of i.i.d. high-dimensional random vectors. To avoid the curse of dimensionality, we impose an additional assumption on the structure of the covariance matrix $Σ$. To be more precise, we study the case when $Σ$ can be approximated by a sum of double Kronecker products of smaller matrices in a tensor train (TT) format. Our setup naturally extends widely known Kronecker sum and CANDECOMP/PARAFAC models but admits richer interaction across modes. We suggest an iterative polynomial time algorithm based on TT-SVD and higher-order orthogonal iteration (HOOI) adapted to Tucker-2 hybrid structure. We derive non-asymptotic dimension-free bounds on the accuracy of covariance estimation taking into account hidden Kronecker product and tensor train structures. The efficiency of our approach is illustrated with numerical experiments. |
| title | Dimension-free Bounds for Covariance Estimation with Tensor-Train Structure |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2510.08174 |