Entropy and Minimax Risk of Hypoelliptic Pseudodifferential Operators
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908912161128448 |
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| author | Allard, Thomas Bölcskei, Helmut |
| author_facet | Allard, Thomas Bölcskei, Helmut |
| contents | We characterize the entropy and minimax risk of a broad class of compact pseudodifferential operators. Under suitable decay and regularity conditions on the symbol, we combine a Weyl-type asymptotic relation between the eigenvalue-counting function and the phase-space volume of the symbol with a general correspondence between spectral quantities, entropy, and minimax risk for compact operators. This approach yields explicit asymptotic formulae for both entropy and minimax risk directly in terms of the symbol. As an application, we derive sharp entropy and minimax risk asymptotics for unit balls in Sobolev spaces on unbounded domains, thereby extending Pinsker's theorem for Sobolev classes beyond the bounded-domain setting, and showing that the sharp asymptotic constants are determined by phase-space geometry rather than domain geometry. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_23744 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Entropy and Minimax Risk of Hypoelliptic Pseudodifferential Operators Allard, Thomas Bölcskei, Helmut Functional Analysis We characterize the entropy and minimax risk of a broad class of compact pseudodifferential operators. Under suitable decay and regularity conditions on the symbol, we combine a Weyl-type asymptotic relation between the eigenvalue-counting function and the phase-space volume of the symbol with a general correspondence between spectral quantities, entropy, and minimax risk for compact operators. This approach yields explicit asymptotic formulae for both entropy and minimax risk directly in terms of the symbol. As an application, we derive sharp entropy and minimax risk asymptotics for unit balls in Sobolev spaces on unbounded domains, thereby extending Pinsker's theorem for Sobolev classes beyond the bounded-domain setting, and showing that the sharp asymptotic constants are determined by phase-space geometry rather than domain geometry. |
| title | Entropy and Minimax Risk of Hypoelliptic Pseudodifferential Operators |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2603.23744 |