Entropy and Minimax Risk of Hypoelliptic Pseudodifferential Operators

TL;DR

This paper provides explicit asymptotic formulas for the entropy and minimax risk of broad classes of compact pseudodifferential operators, linking spectral properties to phase-space geometry and extending classical results to unbounded domains.

Contribution

It introduces a novel approach combining Weyl asymptotics with spectral-entropy relations to derive explicit formulas for entropy and minimax risk in unbounded Sobolev spaces.

Findings

Derived sharp asymptotics for entropy and minimax risk

Extended Pinsker's theorem to unbounded domains

Showed phase-space geometry determines asymptotic constants

Abstract

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.