Uniform Concentration for $α$-subexponential Random Operators
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918380934529024 |
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| author | Diao, Tiankun Hu, Xuanang Ulyanov, Vladimir V. Wang, Hanchao |
| author_facet | Diao, Tiankun Hu, Xuanang Ulyanov, Vladimir V. Wang, Hanchao |
| contents | Random matrices acting on structured sets play a fundamental role in high-dimensional geometry, compressed sensing, and randomized algorithms. Existing results primarily focus on subgaussian models, when random matrices act as near-isometries on sets with optimal tail behavior. Nevertheless, very often in applications we deal with distributions with heavy tails that are not subgaussian but have at least exponential-type tails.
In this work, we study random matrices A whose rows (or columns) have $α$-subexponential tail distributions with $α\in (0,2]$. So subgaussian and sub-exponential models are included in as special cases. We establish concentration type inequality for $Ax$, where x belongs to the bounded subsets of $\mathbb{R}^n$, showing that their geometric distortion is governed by Talagrand's functional of the set and depends on the tail parameter $α$. Our results extend the known optimal inequalities in the subgaussian regime ($α=2$), and provide new guarantees for heavier-tailed, yet exponentially integrable, random matrices.
These findings extend the theory of random matrices beyond the subgaussian framework. Moreover, they yield near-isometric embedding results applicable to dimension reduction and allow us to make robust high-dimensional inference under non-Gaussian measurements. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_09487 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform Concentration for $α$-subexponential Random Operators Diao, Tiankun Hu, Xuanang Ulyanov, Vladimir V. Wang, Hanchao Probability Functional Analysis Random matrices acting on structured sets play a fundamental role in high-dimensional geometry, compressed sensing, and randomized algorithms. Existing results primarily focus on subgaussian models, when random matrices act as near-isometries on sets with optimal tail behavior. Nevertheless, very often in applications we deal with distributions with heavy tails that are not subgaussian but have at least exponential-type tails. In this work, we study random matrices A whose rows (or columns) have $α$-subexponential tail distributions with $α\in (0,2]$. So subgaussian and sub-exponential models are included in as special cases. We establish concentration type inequality for $Ax$, where x belongs to the bounded subsets of $\mathbb{R}^n$, showing that their geometric distortion is governed by Talagrand's functional of the set and depends on the tail parameter $α$. Our results extend the known optimal inequalities in the subgaussian regime ($α=2$), and provide new guarantees for heavier-tailed, yet exponentially integrable, random matrices. These findings extend the theory of random matrices beyond the subgaussian framework. Moreover, they yield near-isometric embedding results applicable to dimension reduction and allow us to make robust high-dimensional inference under non-Gaussian measurements. |
| title | Uniform Concentration for $α$-subexponential Random Operators |
| topic | Probability Functional Analysis |
| url | https://arxiv.org/abs/2603.09487 |