Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run

What the math gives to ML

The paper gives a quantitative way to turn uniform sampling from a convex body into a rapidly mixing Markov kernel, with a spectral-gap bound controlled by the target distribution's Poincare constant rather than only its outer radius. The transferable asset is the dimension-explicit contraction guarantee for line-based, constraint-preserving random moves, including logarithmic dependence on the initial chi-squared distance. A plausible neural-network use is to sample adversarial, augmentation, or latent perturbations from convex feasible sets without rejection sampling, especially after whitening or rounding the set so its covariance is well conditioned. This is not a generic optimizer for nonconvex weights; its strongest application is uniform exploration of convex perturbation domains embedded inside a training or robustness pipeline.