Rank-Based Norms, Capra-Conjugacies and the Rank Function

TL;DR

This paper introduces a new coupling called Capra for matrices, computes the conjugates of the rank function using rank-based norms, and provides a variational formula that characterizes the rank function, especially for the Frobenius norm.

Contribution

It defines the Capra conjugate and biconjugate of the rank function in terms of generalized r-rank norms, offering new insights into matrix rank analysis.

Findings

Capra conjugate of the rank function expressed via rank-based norms

Variational lower bound for the rank function derived

Exact characterization of the rank function for Frobenius norm

Abstract

We consider the space of matrices, with given number of rows and of columns, equipped with the classic trace scalar product. With any matrix (source) norm, we associate a coupling, called Capra, between the space of matrices and itself. Then, we compute the Capra conjugate and biconjugate of the rank function. They are expressed in function of a sequence of rank-based norms, more precisely generalized r-rank and dual r-rank matrix norms associated with the matrix source norm. We deduce a lower bound of the rank function given by a variational formula which involves the generalized r-rank norms. In the case of the Frobenius norm, we show that the rank function is equal to the variational formula.