Upper bounds for the L^q empirical process via generic chaining
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909894292013056 |
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| author | Shang, Zong |
| author_facet | Shang, Zong |
| contents | Using the generic chaining method, we derive upper bounds for the \(L^q\) process of sub-Gaussian classes when \(1 \le q \le 2\), thereby resolving an open problem posed by Al-Ghattas, Chen, and Sanz-Alonso in arXiv:2502.16916. Combined with the results of arXiv:2502.16916, this yields upper bounds for the \(L^q\) process for all \(1 \le q < \infty\). We also present corollaries of this result in the geometry of Banach spaces, including high-probability bounds on the \(\ell_q\) norm diameter of random hyperplane sections of convex bodies where the subspaces are not necessarily uniformly distributed on the Grassmannian manifold and the restricted isomorphic property for \(\ell_q\) norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06338 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Upper bounds for the L^q empirical process via generic chaining Shang, Zong Probability Using the generic chaining method, we derive upper bounds for the \(L^q\) process of sub-Gaussian classes when \(1 \le q \le 2\), thereby resolving an open problem posed by Al-Ghattas, Chen, and Sanz-Alonso in arXiv:2502.16916. Combined with the results of arXiv:2502.16916, this yields upper bounds for the \(L^q\) process for all \(1 \le q < \infty\). We also present corollaries of this result in the geometry of Banach spaces, including high-probability bounds on the \(\ell_q\) norm diameter of random hyperplane sections of convex bodies where the subspaces are not necessarily uniformly distributed on the Grassmannian manifold and the restricted isomorphic property for \(\ell_q\) norm. |
| title | Upper bounds for the L^q empirical process via generic chaining |
| topic | Probability |
| url | https://arxiv.org/abs/2511.06338 |