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SUMMARY:Complexity of sampling truncated log-concave measures\, and the ro
 le of stochastic localization - Yuansi Chen (ETH Zurich)
DTSTART:20250606T130000Z
DTEND:20250606T140000Z
UID:TALK231523@talks.cam.ac.uk
CONTACT:Qingyuan Zhao
DESCRIPTION:Motivated by computational challenges in Bayesian models with 
 indicator variables\, such as probit/tobit regression\, we study the compu
 tational complexity of drawing samples from a truncated log-concave measur
 e. We discuss two problems. In the first part\, using stochastic localizat
 ion as a way to reduce the sampling problem to truncated Gaussians\, we an
 alyze the hit-and-run algorithm for sampling uniformly from an isotropic c
 onvex body in n dimensions and establish $n^2$ mixing time. In the second 
 part\, building on interior point methods\, we analyze the mixing time of 
 regularized Dikin walks for sampling log-concave measures truncated on a p
 olytope. For a logconcave and log-smooth distribution with condition numbe
 r $\\kappa$\, truncated on a polytope in $R^n$ defined with $m$ linear con
 straints\, we prove that the soft-threshold Dikin walk mixes in $O((m+\\ka
 ppa)n)$ iterations from a warm initialization. It improves upon prior work
  which required the polytope to be bounded and involved a bound dependent 
 on the radius of the bounded region. Here\, stochastic localization allows
  us to extend the analysis to weakly log-concave measures. \nhttps://arxiv
 .org/abs/2212.00297\nhttps://arxiv.org/abs/2412.11303
LOCATION:MR12\, Centre for Mathematical Sciences
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