John's walk

We present an affine-invariant random walk for drawing uniform random samples from a convex body K subset of R-n that uses maximum-volume inscribed ellipsoids, known as John's ellipsoids, for the proposal distribution. Our algorithm makes steps using uniform sampling from the John's ellipsoid of the symmetrization of K at the current point. We show that from a warm start, the random walk mixes in (O) over tilde (n(7)) steps, where the log factors hidden in the (O) over tilde depend only on constants associated with the warm start and desired total variation distance to uniformity. We also prove polynomial mixing bounds starting from any fixed point x such that for any chord pq of K containing x, vertical bar log vertical bar p-x vertical bar/vertical bar q-x vertical bar vertical bar is bounded above by a polynomial in n.