The Geometry of Efficient Nonconvex Sampling

We present an efficient algorithm for uniformly sampling from an arbitrary compact body $\mathcal{X}\subset {\mathbb{R}}^n$ from a warm start under isoperimetry and a natural volume growth condition. Our result provides a substantial common generalization of known results for convex bodies and star-shaped bodies. The complexity of the algorithm is polynomial in the dimension, the Poincaré constant of the uniform distribution on $\mathcal{X}$ and the volume growth constant of the set $\mathcal{X}$.