On the Injective Norm of Sums of Random Tensors and the Moments of Gaussian Chaoses
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916651907153920 |
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| author | Aden-Ali, Ishaq |
| author_facet | Aden-Ali, Ishaq |
| contents | We prove an upper bound on the expected $\ell_p$ injective norm of sums of subgaussian random tensors. Our proof is simple and does not rely on any explicit geometric or chaining arguments. Instead, it follows from a simple application of the PAC-Bayesian lemma, a tool that has proven effective at controlling the suprema of certain ``smooth'' empirical processes in recent years. Our bound strictly improves a very recent result of Bandeira, Gopi, Jiang, Lucca, and Rothvoss. In the Euclidean case ($p=2$), our bound sharpens a result of Latała that was central to proving his estimates on the moments of Gaussian chaoses. As a consequence, we obtain an elementary proof of this fundamental result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10580 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Injective Norm of Sums of Random Tensors and the Moments of Gaussian Chaoses Aden-Ali, Ishaq Probability Machine Learning Statistics Theory We prove an upper bound on the expected $\ell_p$ injective norm of sums of subgaussian random tensors. Our proof is simple and does not rely on any explicit geometric or chaining arguments. Instead, it follows from a simple application of the PAC-Bayesian lemma, a tool that has proven effective at controlling the suprema of certain ``smooth'' empirical processes in recent years. Our bound strictly improves a very recent result of Bandeira, Gopi, Jiang, Lucca, and Rothvoss. In the Euclidean case ($p=2$), our bound sharpens a result of Latała that was central to proving his estimates on the moments of Gaussian chaoses. As a consequence, we obtain an elementary proof of this fundamental result. |
| title | On the Injective Norm of Sums of Random Tensors and the Moments of Gaussian Chaoses |
| topic | Probability Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2503.10580 |