Concentration inequalities for random tensors
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866918153879027712 |
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| author | Vershynin, Roman |
| author_facet | Vershynin, Roman |
| contents | We show how to extend several basic concentration inequalities for simple random tensors $X = x_1 \otimes \cdots \otimes x_d$ where all $x_k$ are independent random vectors in $\mathbb{R}^n$ with independent coefficients. The new results have optimal dependence on the dimension $n$ and the degree $d$. As an application, we show that random tensors are well conditioned: $(1-o(1)) n^d$ independent copies of the simple random tensor $X \in \mathbb{R}^{n^d}$ are far from being linearly dependent with high probability. We prove this fact for any degree $d = o(\sqrt{n/\log n})$ and conjecture that it is true for any $d = O(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1905_00802 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Concentration inequalities for random tensors Vershynin, Roman Probability 60F10, 60B20 We show how to extend several basic concentration inequalities for simple random tensors $X = x_1 \otimes \cdots \otimes x_d$ where all $x_k$ are independent random vectors in $\mathbb{R}^n$ with independent coefficients. The new results have optimal dependence on the dimension $n$ and the degree $d$. As an application, we show that random tensors are well conditioned: $(1-o(1)) n^d$ independent copies of the simple random tensor $X \in \mathbb{R}^{n^d}$ are far from being linearly dependent with high probability. We prove this fact for any degree $d = o(\sqrt{n/\log n})$ and conjecture that it is true for any $d = O(n)$. |
| title | Concentration inequalities for random tensors |
| topic | Probability 60F10, 60B20 |
| url | https://arxiv.org/abs/1905.00802 |