John's Walk
This provides a polynomial-time algorithm for uniform sampling from convex bodies, addressing a fundamental challenge in computational geometry and optimization.
The paper tackles the problem of uniformly sampling from convex bodies in high dimensions by introducing an affine-invariant random walk based on John's ellipsoids, achieving a mixing time of O(n^7) from a warm start.
We present an affine-invariant random walk for drawing uniform random samples from a convex body $\mathcal{K} \subset \mathbb{R}^n$ 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 $\mathcal{K}$ at the current point. We show that from a warm start, the random walk mixes in $\widetilde{O}(n^7)$ steps where the log factors 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 $\mathcal{K}$ containing $x$, $\left|\log \frac{|p-x|}{|q-x|}\right|$ is bounded above by a polynomial in $n$.