Joseph Lehec

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

We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected stochastic gradient descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main result shows in particular that when the target distribution is uniform, LMC mixes in

Moments of the Gaussian Chaos

\widetilde{O}(n^7)

Borell's formula on a Riemannian manifold and applications

O~(n7) steps (where n is the dimension). We also provide preliminary experimental evidence that LMC performs at least as well as hit-and-run, for which a better mixing time of

Functional Versions of Lp-Affine Surface Area and Entropy Inequalities

\widetilde{O}(n^4)

Regularization in L_1 for the Ornstein-Uhlenbeck semigroup

O~(n4) was proved by Lovász and Vempala.

Poisson processes and a log-concave Bernstein theorem

Chaining techniques show that if X is an isotropic log-concave random vector in \(\mathbb{R}^{n}\) and Γ is a standard Gaussian vector then