Concentration inequalities for smooth random fields

TL;DR

This paper establishes a sharp concentration inequality for the supremum of smooth random fields, linking it to intrinsic problem dimension, with applications to eigenvalue-related functions of random matrices.

Contribution

It introduces a novel concentration inequality for smooth random fields, connecting supremum bounds to intrinsic problem dimensions, and applies it to eigenvalue functions of random matrices.

Findings

Supremum of smooth random fields can be bounded with high probability.

The inequality applies to functions of maximal eigenvalues of random matrices.

Provides a new tool for analyzing high-dimensional random processes.

Abstract

In this note we derive a sharp concentration inequality for the supremum of a smooth random field over a finite dimensional set. It is shown that this supremum can be bounded with high probability by the value of the field at some deterministic point plus an intrinsic dimension of the optimisation problem. As an application we prove the exponential inequality for a function of the maximal eigenvalue of a random matrix is proved.