arXiv subjects

Oisín Faust

Publications and source records attributed to Oisín Faust.

2 recordsLinked to original sources

arXiv2023

Rational approximations of operator monotone and operator convex functions

Operator convex functions defined on the positive half-line play a prominent role in the theory of quantum information, where they are used to define quantum $f$-divergences. Such functions admit integral representations in terms of rational functions. Obtaining high-quality rational approximants of operator convex functions is particularly useful for solving optimization problems involving quantum $f$-divergences using semidefinite programming. In this paper we study the quality of rational approximations of operator convex (and operator monotone) functions. Our main theoretical results are precise global bounds on the error of local Padé-like approximants, as well as minimax approximants, with respect to different weight functions. While the error of Padé-like approximants depends inverse polynomially on the degree of the approximant, the error of minimax approximants has root exponential dependence and we give detailed estimates of the exponents in both cases. We also explain how minimax approximants can be obtained in practice using the differential correction algorithm.

math.OC↗

arXiv2022

Sum-of-Squares proofs of logarithmic Sobolev inequalities on finite Markov chains

Logarithmic Sobolev inequalities are a fundamental class of inequalities that play an important role in information theory. They play a key role in establishing concentration inequalities and in obtaining quantitative estimates on the convergence to equilibrium of Markov processes. More recently, deep links have been established between logarithmic Sobolev inequalities and strong data processing inequalities. In this paper we study logarithmic Sobolev inequalities from a computational point of view. We describe a hierarchy of semidefinite programming relaxations which give certified lower bounds on the logarithmic Sobolev constant of a finite Markov operator, and we prove that the optimal values of these semidefinite programs converge to the logarithmic Sobolev constant. Numerical experiments show that these relaxations are often very close to the true constant even for low levels of the hierarchy. Finally, we exploit our relaxation to obtain a sum-of-squares proof that the logarithmic Sobolev constant is equal to half the Poincaré constant for the specific case of a simple random walk on the odd $n$-cycle, with $n\in\{5,7,\dots,21\}$. Previously this was known only for $n=5$ and even $n$.

math.OC↗