A Matrix Generalization of the Hardy-Littlewood-P\'olya Rearrangement Inequality and Its Applications

TL;DR

This paper generalizes the Hardy-Littlewood-Pólya rearrangement inequality to positive definite and rectangular matrices, providing new inequalities applicable in signal processing and machine learning.

Contribution

It introduces a matrix-based generalization of a classical inequality and extends it to rectangular matrices, with applications in various computational fields.

Findings

Generalized rearrangement inequality for positive definite matrices

Extended the inequality to rectangular matrices

Derived new inequalities for distance-like functions

Abstract

By analyzing an optimization problem over orthogonal matrices, we prove a generalization of the Hardy-Littlewood-P\'olya rearrangement inequality to positive definite matrices. The inequality is then extended to rectangular matrices. Using our main results, we derive new inequalities for several distance-like functions encountered in various signal processing or machine learning applications.