Robust Eigenvectors of Symmetric Tensors
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866916663639670784 |
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| author | Muller, Tommi Robeva, Elina Usevich, Konstantin |
| author_facet | Muller, Tommi Robeva, Elina Usevich, Konstantin |
| contents | The tensor power method generalizes the matrix power method to higher order arrays, or tensors. Like in the matrix case, the fixed points of the tensor power method are the eigenvectors of the tensor. While every real symmetric matrix has an eigendecomposition, the vectors generating a symmetric decomposition of a real symmetric tensor are not always eigenvectors of the tensor.
In this paper we show that whenever an eigenvector is a generator of the symmetric decomposition of a symmetric tensor, then (if the order of the tensor is sufficiently high) this eigenvector is robust, i.e., it is an attracting fixed point of the tensor power method. We exhibit new classes of symmetric tensors whose symmetric decomposition consists of eigenvectors. Generalizing orthogonally decomposable tensors, we consider equiangular tight frame decomposable and equiangular set decomposable tensors. Our main result implies that such tensors can be decomposed using the tensor power method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_06880 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Robust Eigenvectors of Symmetric Tensors Muller, Tommi Robeva, Elina Usevich, Konstantin Numerical Analysis Algebraic Geometry Spectral Theory 15A69, 15A18, 42C15, 65F15 The tensor power method generalizes the matrix power method to higher order arrays, or tensors. Like in the matrix case, the fixed points of the tensor power method are the eigenvectors of the tensor. While every real symmetric matrix has an eigendecomposition, the vectors generating a symmetric decomposition of a real symmetric tensor are not always eigenvectors of the tensor. In this paper we show that whenever an eigenvector is a generator of the symmetric decomposition of a symmetric tensor, then (if the order of the tensor is sufficiently high) this eigenvector is robust, i.e., it is an attracting fixed point of the tensor power method. We exhibit new classes of symmetric tensors whose symmetric decomposition consists of eigenvectors. Generalizing orthogonally decomposable tensors, we consider equiangular tight frame decomposable and equiangular set decomposable tensors. Our main result implies that such tensors can be decomposed using the tensor power method. |
| title | Robust Eigenvectors of Symmetric Tensors |
| topic | Numerical Analysis Algebraic Geometry Spectral Theory 15A69, 15A18, 42C15, 65F15 |
| url | https://arxiv.org/abs/2111.06880 |