In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies

arXiv:2405.01425v4 Announce Type: replace-cross
Abstract: We present a new random walk for uniformly sampling high-dimensional convex bodies. It achieves state-of-the-art runtime complexity with stronger guarantees on the output than previously known, namely in R’enyi divergence (which implies TV, $mathcal{W}_2$, KL, $chi^2$). The proof departs from known approaches for polytime algorithms for the problem — we utilize a stochastic diffusion perspective to show contraction to the target distribution with the rate of convergence determined by functional isoperimetric constants of the target distribution.