Small Ball Probabilities for Simple Random Tensors
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929296226910208 |
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| author | Hu, Xuehan Paouris, Grigoris |
| author_facet | Hu, Xuehan Paouris, Grigoris |
| contents | We study the small ball probability of an order-$\ell$ simple random tensor $X=X^{(1)}\otimes\cdots\otimes X^{(\ell)}$ where $X^{(i)}, 1\leq i\leq\ell$ are independent random vectors in $\mathbb{R}^n$ that are log-concave or have independent coordinates with bounded densities. We show that the probability that the projection of $X$ onto an $m$-dimensional subspace $F$ falls within an Euclidean ball of length $\varepsilon$ is upper bounded by $\frac{\varepsilon}{(\ell-1)!}\left(C\log\left(\frac{e}{\varepsilon}\right)\right)^{\ell}$ and also this upper bound is sharp when $m$ is small. We also established that a much better estimate holds true for a random subspace. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_20192 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small Ball Probabilities for Simple Random Tensors Hu, Xuehan Paouris, Grigoris Probability Functional Analysis We study the small ball probability of an order-$\ell$ simple random tensor $X=X^{(1)}\otimes\cdots\otimes X^{(\ell)}$ where $X^{(i)}, 1\leq i\leq\ell$ are independent random vectors in $\mathbb{R}^n$ that are log-concave or have independent coordinates with bounded densities. We show that the probability that the projection of $X$ onto an $m$-dimensional subspace $F$ falls within an Euclidean ball of length $\varepsilon$ is upper bounded by $\frac{\varepsilon}{(\ell-1)!}\left(C\log\left(\frac{e}{\varepsilon}\right)\right)^{\ell}$ and also this upper bound is sharp when $m$ is small. We also established that a much better estimate holds true for a random subspace. |
| title | Small Ball Probabilities for Simple Random Tensors |
| topic | Probability Functional Analysis |
| url | https://arxiv.org/abs/2403.20192 |