06.11.2024, 14:00 - 16:00
– Campus Golm, Building 9, Room 2.22 and via Zoom
Institutskolloquium
Graphon Models for Inhomogeneous Random Graphs
Olga Klopp (Paris), Nicolas Verzelen (Montpellier)
Sven Wang (HU Berlin)
We consider the problem of generating random samples from high-dimensional posterior distributions. We will discuss both (i) conditions under which diffusion-based MCMC algorithms mix in polynomial-time (based on https://arxiv.org/pdf/2009.05298.pdf) as well as (ii) situations in which `cold-start' MCMC suffers from an exponentially long mixing time (based on https://arxiv.org/pdf/2209.02001.pdf). We will focus on the setting of non-linear inverse regression models. Our positive results on polynomial-time mixing are derived under local `gradient stability' assumptions on the forward map, which can be verified for a range of well-known non-linear inverse problems involving elliptic PDE, as well as under the assumption that a good initializer is available. Our negative results on exponentially long mixing times hold for `cold-start' MCMC. We show that there exist non-linear regression models in which the posterior distribution is unimodal, but there exists a so-called `free entropy barrier', which local Markov chains take an exponentially long time to traverse.