Computing the norm of nonnegative matrices and the log-Sobolev constant of Markov chains

TL;DR

This paper develops new convergence guarantees for computing nonnegative matrix norms, including NP-hard cases, and uses these results to establish a lower bound on the log-Sobolev constant of Markov chains.

Contribution

It introduces a global convergence theorem for nonnegative matrices and applies it to derive bounds on the log-Sobolev constant of Markov chains.

Findings

New convergence guarantees for mixed-subordinate matrix norms

Explicit bounds for NP-hard matrix norm computations

Lower bound on the log-Sobolev constant of Markov chains

Abstract

We analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $\ell^p$ matrix norms. In particular, exploiting the Birkoff--Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norms induced by the vector norms made by the sum of different $\ell^p$-norms of subsets of entries. Finally, we use the new results combined with hypercontractive inequalities to prove a new lower bound on the logarithmic Sobolev constant of a Markov chain.