Hit-and-Run Mixes as Fast as the Ball Walk

What the math gives to ML

The paper gives a constructive sampling method with a strong warm-start guarantee: hit-and-run reaches near-uniformity on an isotropic convex body with only logarithmic dependence on the warmness and accuracy parameters. The transferable asset is not merely hit-and-run itself, but the fact that a chain initialized from a cheap approximate sampler can rapidly correct its bias without the polynomial burn-in previously required under the same dimension dependence. A practical neural-network use is constrained latent or feature augmentation: whiten a convex feasible region, initialize from a simple interior distribution, and use hit-and-run to generate diverse approximately uniform training examples or adversarial probes.