Aggregating regular norms

TL;DR

This paper studies how to combine regular norms on finite-dimensional spaces while maintaining their regularity properties, which are crucial for applications in convex geometry, probability, and optimization.

Contribution

It introduces methods for aggregating regular norms with controlled inflation of regularity parameters, enabling dimension-independent geometric and optimization results.

Findings

Provides aggregation techniques for regular norms

Ensures controlled regularity inflation during aggregation

Facilitates dimension-independent applications in geometry and optimization

Abstract

The subject of this paper is regularity-preserving aggregation of regular norms on finite-dimensional linear spaces. Regular norms were introduced in [5] and are closely related to ``type 2'' spaces [9, Chapter 9] playing important role in 1) high-dimensional convex geometry and probability in Banach spaces [0.9.12.13.15], and in 2) design of proximal first-order algorithms for large-scale convex optimization with dimension-independent, or nearly so, complexity. Regularity, with moderate parameters, of a norm makes applicable, in a dimension-independent fashion, numerous geometric, probabilistic, and optimization-related results, which motivates our interest in aggregating regular norms with controlled (and moderate) inflation of regularity parameters.