Optimal Sparsification of Gaussian Processes

2026-06-17

AI Summary

This work addresses the problem of efficiently approximating the supremum of a centered Gaussian process in the L² norm without dependence on the ambient dimension or the size of the original index set. By constructing a shifted sub-process supported on only exp(O(1/ε²)) points, the authors achieve an approximation error bounded by ε times the Gaussian width, significantly improving upon prior results. They further establish that this exponential dependence on ε is tight. The key technical ingredients combine Sudakov’s minoration with the Brascamp–Lieb inequality in an interpolation argument. As a corollary, they derive an exponentially improved junta theorem for norms in Gaussian space, yielding stronger applications in learning theory, property testing, and polyhedral approximation of convex bodies.

Abstract

We prove an optimal dimension-free sparsification theorem for suprema of centered Gaussian processes. Given a bounded set $T\subseteq\mathbb{R}^n$, we show that the supremum of the canonical Gaussian process on $T$ can be $L^2$-approximated by the supremum of a shifted subprocess indexed by only $\exp(O(1/\varepsilon^2))$ points, with error at most $\varepsilon$ times the Gaussian width of $T$. In particular, the size of the approximating process is independent of both the ambient dimension and the cardinality of the original index set. This improves a recent sparsification theorem of De, Nadimpalli, O'Donnell, and Servedio (2026) by an exponential factor, and we show that the dependence on $\varepsilon$ is tight up to constants in the exponent. As consequences, we obtain an exponentially improved junta theorem for norms over Gaussian space and sharpen results on learning, property testing, and polyhedral approximation of convex sets under the Gaussian measure. The proof is based on an interpolation argument that combines Sudakov's minoration with the Brascamp--Lieb inequality.

Problem

• Gaussian processes
• sparsification
• suprema
• dimension-free
• Gaussian width

Innovation

• Gaussian processes
• sparsification
• dimension-free
• Gaussian width
• Brascamp–Lieb inequality