Ellipsoid methods for metric entropy computation

Authors

Thomas Allard and Helmut Bölcskei

Reference

Constructive Approximation, revised version, Dec. 2025, submitted.

Abstract

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. From a broader perspective, our framework can be seen as a step toward using metric entropy not only at an order-of-magnitude level but also in a more quantitatively refined way for complexity estimation.

Keywords

Metric entropy, ellipsoids, complex analysis

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