Gaussian Width of Convex Sets via Integral Decompositions, Projections, and the Distribution of Intrinsic Volumes

2026-03-03

AI Summary

This study addresses the problem of effectively characterizing and upper-bounding the expected supremum of Gaussian processes indexed by convex sets—commonly known as Gaussian width. The work proposes two novel decomposition strategies that circumvent traditional chaining constructions: one based on metric projections of Gaussian vectors onto scaled versions of the convex set, and another derived from a fixed point of a quadratically penalized variant of local widths. By establishing a new connection between Gaussian width and the distribution of intrinsic volumes of the convex set, the analysis reveals that the Gaussian width is governed by a single “peak index” in the sequence of intrinsic volumes. In the worst case, this framework recovers a localized form of Dudley’s integral. Consequently, controlling Gaussian width reduces to analyzing the peak behavior of intrinsic volumes, yielding sharper and more geometrically intuitive upper bounds.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Security and Privacy: Large-scale security measurements

Abstract

We revisit the problem of bounding the expected supremum of a canonical Gaussian process indexed by a convex set $T \subset \mathbf{R}^d$. We develop two decompositions for the Gaussian width, based on the geometry of the index set. The first decomposition involves metric projections of Gaussians onto rescaled copies of $T$. The second involves fixed points arising from a quadratically penalized variant of the local width. Neither decomposition directly invokes generic chaining constructions. Our results make use of recent work in geometric analysis and Gaussian processes. The work of Chatterjee [Ann. Statist., 2014] characterizes the behavior of the metric projection of a Gaussian random vector onto rescaled copies of $T$ with a variational problem involving localized Gaussian widths. We use these bounds to develop decompositions of the Gaussian width using the local metric structure of $T$. Second, we leverage the work of Vitale [Ann. Probab., 1996] to form a connection between the Wills functional (and hence the intrinsic volumes of $T$) and the first terms that appear in our decompositions. Finally, invoking recent work by Mourtada [J. Eur. Math. Soc., 2025] on the logarithm of the Wills functional, we show that the width is controlled by a single, ''peak index'' of the intrinsic volumes. In the worst case, our bound recovers a local form of the classical Dudley integral.

Problem

• Gaussian width
• convex sets
• intrinsic volumes
• Gaussian processes
• metric projections

Innovation

• Gaussian width
• intrinsic volumes
• metric projection
• Wills functional
• local width