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Stein Points
An important task in computational statistics and machine learning is to approximate a posterior distribution with an empirical measure supported on a set of representative points . This paper focuses on methods where the selection of points is essentially deterministic, with an emphasis on achieving accurate approximation when is small. To this end, we present `Stein Points'. The idea is to exploit either a greedy or a conditional gradient method to iteratively minimise a kernel Stein discrepancy between the empirical measure and . Our empirical results demonstrate that Stein Points enable accurate approximation of the posterior at modest computational cost. In addition, theoretical results are provided to establish convergence of the method
Noncritical holomorphic functions on Stein spaces
We prove that every reduced Stein space admits a holomorphic function without
critical points. Furthermore, any closed discrete subset of such a space is the
critical locus of a holomorphic function. We also show that for every complex
analytic stratification with nonsingular strata on a reduced Stein space there
exists a holomorphic function whose restriction to every stratum is
noncritical. These result also provide some information on critical loci of
holomorphic functions on desingularizations of Stein spaces. In particular,
every 1-convex manifold admits a holomorphic function that is noncritical
outside the exceptional variety.Comment: To appear in J. Eur. Math. Soc. (JEMS
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