Computing the norm of nonnegative matrices and the log-Sobolev constant of Markov chains
share
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 ℓ^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 ℓ^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.
READ FULL TEXT