We review and formulate results concerning log-concavity and strong-log-concavity in both
discrete and continuous settings. We show how preservation of log-concavity and strongly …

1.1. 3 A simple open problem. 1.2 The Perron-Frobenius Theorem.... 1.2. 1 Two proofs of the
Perron-Frobenius theorem 1.2. 2 Comments on the Perron-Frobenius theorem 1.2. 3 Further …

Diffusion models are a powerful method for generating approximate samples from high-
dimensional data distributions. Several recent results have provided polynomial bounds on …

Classically, the continuous-time Langevin diffusion converges exponentially fast to its
stationary distribution π under the sole assumption that π satisfies a Poincaré inequality …

Heuristically, a discrete-time stochastic process has the Markov property if the past and
future are independent given the present. In this introductory chapter, we give the formal …

Parameter studies are everywhere in computational science. Complex engineering
simulations must run several times with different inputs to effectively study the relationships …

We study the problem of sampling from a strongly log-concave density supported on
$\mathbb {R}^ d $, and prove a non-asymptotic upper bound on the mixing time of the …

It is a fundamental problem to understand the complexity of high-accuracy sampling from a
strongly log-concave density π on ℝ d. Indeed, in practice, high-accuracy samplers such as …

This book is a continuation of Asymptotic Geometric Analysis, Part I, which was published as
volume 202 in this series. Asymptotic geometric analysis studies properties of geometric …

Optimization algorithms and Monte Carlo sampling algorithms have provided the
computational foundations for the rapid growth in applications of statistical machine learning …