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We extend the Langevin Monte Carlo (LMC) algorithm to compactly supportedmeasures via a projection step, akin to projected Stochastic Gradient Descent(SGD). We show that (projected) LMC allows to sample in polynomial time from alog-concave distribution with smooth potential. This gives a new Markov chainto sample from a log-concave distribution. Our main result shows in particularthat when the target distribution is uniform, LMC mixes in $\tilde{O}(n^7)$steps (where $n$ is the dimension). We also provide preliminary experimentalevidence that LMC performs at least as well as hit-and-run, for which a bettermixing time of $\tilde{O}(n^4)$ was proved by Lov{\'a}sz and Vempala.

Sampling from a log-concave distribution with Projected Langevin Monte Carlo