Regular functional covering numbers

TL;DR

This paper proves the existence of a regular functional M-position for geometric log-concave functions, enabling uniform control of covering numbers across scales, extending Pisier's concepts from convex bodies to functions.

Contribution

It introduces a functional analogue of Pisier's regular M-positions for log-concave functions, providing a new framework for analyzing their geometric properties.

Findings

Establishes a regular functional M-position for isotropic geometric log-concave functions.

Provides bounds on covering numbers at all scales for these functions.

Shows the isotropic position yields an almost 1-regular functional M-position.

Abstract

We establish the existence of a regular functional $M$-position, in the sense of Pisier, for geometric log-concave functions. This provides a functional analogue of Pisier's regular $M$-positions for convex bodies and yields uniform control of covering numbers at all scales. Specifically, we show that every isotropic geometric log-concave function $f:\mathbb{R}^n \to [0,\infty)$ satisfies, for all $t\geq 1$, $$\max \left\{N(f, t \cdot g),\,N(f^*, t \cdot g),\,N(g, t \cdot f),\,N(g, t \cdot f^*)\right\} \leq \exp\left( \frac{\gamma_n^2\, n}{t} \right),$$ where $f^*$ denotes the Legendre dual of $f$, $(t \cdot f)(x)=f(x/t)$ is the $t$-homothety of $f$, $g(x)=\exp \left(-\frac{1}{2}|x|^{2}\right)$ and $\gamma_n \leq c(\ln n)^2$. Our result shows that the isotropic position of a log-concave function already provides an almost $1$-regular functional $M$-position.