Searcharxiv⌕ Search

arXiv subjects

Thomas Allard

Publications and source records attributed to Thomas Allard.

6 recordsLinked to original sources
arXiv2026

Entropy and Minimax Risk of Hypoelliptic Pseudodifferential Operators

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.

math.FA↗
arXiv2025

Metric Entropy and Minimax Risk of Ellipsoids with an Application to Pinsker's Theorem

We study how large an $\ell^2$ ellipsoid is by introducing type-$\tau$ integrals that capture the average decay of its semi-axes. These integrals turn out to be closely related to standard complexity measures: we show that the metric entropy of the ellipsoid is asymptotically equivalent to the type-1 integral, and that the minimax risk in non-parametric estimation is asymptotically determined by the type-2 and type-3 integrals. This allows us to retrieve and sharpen classical results about metric entropy and minimax risk of ellipsoids through a systematic analysis of the type-$\tau$ integrals, and yields an explicit formula linking the two. As an application, we improve on the best-known characterization of the metric entropy of the Sobolev ellipsoid, and extend Pinsker's Sobolev theorem in two ways: (i) to any bounded open domain in arbitrary finite dimension, and (ii) by providing the second-order term in the asymptotic expansion of the minimax risk.

math.ST↗
arXiv2025

Metric Entropy of Ellipsoids in Banach Spaces: Techniques and Precise Asymptotics

We develop new techniques for computing the metric entropy of ellipsoids -- with polynomially decaying semi-axes -- in Banach spaces. Besides leading to a unified and comprehensive framework, these tools deliver numerous novel results as well as substantial improvements and generalizations of classical results. Specifically, we characterize the constant in the leading term in the asymptotic expansion of the metric entropy of $p$-ellipsoids with respect to $q$-norm, for arbitrary $p,q \in [1, \infty]$, to date known only in the case $p=q=2$. Moreover, for $p=q=2$, we improve upon classical results by specifying the second-order term in the asymptotic expansion. In the case $p=q=\infty$, we obtain a complete, as opposed to asymptotic, characterization of metric entropy and explicitly construct optimal coverings. To the best of our knowledge, this is the first exact characterization of the metric entropy of an infinite-dimensional body. Application of our general results to function classes yields an improvement of the asymptotic expansion of the metric entropy of unit balls in Sobolev spaces and identifies the dependency of the metric entropy of unit balls in Besov spaces on the domain of the functions in the class. Sharp results on the metric entropy of function classes find application, e.g., in machine learning, where they allow to specify the minimum required size of deep neural networks for function approximation, nonparametric regression, and classification over these function classes.

math.FA↗
arXiv2024

Metric Entropy of Analytic Function Classes via Ellipsoidal Methods

We present a systematic methodology for characterizing the metric entropy of infinite-dimensional ellipsoids with exponentially decaying semi-axes. The approach does not rely on the explicit construction of coverings or packings and yields a unified framework for deriving sharp entropy estimates for a wide range of analytic function classes, including periodic functions analytic on a strip, analytic functions bounded on a disk, and functions of exponential type. In each of these cases, our results improve upon the best known bounds in the literature.

math.FA↗
arXiv2022

Metric entropy of causal, discrete-time LTI systems

In [1] it is shown that recurrent neural networks (RNNs) can learn - in a metric entropy optimal manner - discrete time, linear time-invariant (LTI) systems. This is effected by comparing the number of bits needed to encode the approximating RNN to the metric entropy of the class of LTI systems under consideration [2, 3]. The purpose of this note is to provide an elementary self-contained proof of the metric entropy results in [2, 3], in the process of which minor mathematical issues appearing in [2, 3] are cleaned up. These corrections also lead to the correction of a constant in a result in [1] (see Remark 2.5).

math.DS↗