Chaining, Interpolation, and Convexity II: The contraction principle

TL;DR

This paper introduces a refined interpolation-based contraction principle that simplifies controlling chaining functionals, leading to new bounds on random matrices and a streamlined proof of the majorizing measure theorem.

Contribution

It presents a new contraction principle that enhances the interpolation method for bounding chaining functionals, improving efficiency and sharpness of results.

Findings

New dimension-free bounds on norms of random matrices

Simplified proof of the majorizing measure theorem

Enhanced control of chaining functionals via interpolation

Abstract

The generic chaining method provides a sharp description of the suprema of many random processes in terms of the geometry of their index sets. The chaining functionals that arise in this theory are however notoriously difficult to control in any given situation. In the first paper in this series, we introduced a particularly simple method for producing the requisite multi scale geometry by means of real interpolation. This method is easy to use, but does not always yield sharp bounds on chaining functionals. In the present paper, we show that a refinement of the interpolation method provides a canonical mechanism for controlling chaining functionals. The key innovation is a simple but powerful contraction principle that makes it possible to efficiently exploit interpolation. We illustrate the utility of this approach by developing new dimension-free bounds on the norms of random matrices and on chaining functionals in Banach lattices. As another application, we give a remarkably short interpolation proof of the majorizing measure theorem that entirely avoids the greedy construction that lies at the heart of earlier proofs.