Warm-start hit-and-run augmentation

1. Integration point: add a CPU or GPU sampler before the data-loader collate step, or use it as a latent-augmentation module for a VAE or diffusion model. Define a convex feasible region K in feature or latent coordinates, such as the intersection of box constraints, linear safety constraints, and a trust-region polytope. The sampler outputs augmented vectors that are decoded or fed directly to the network. 2. Pseudocode: estimate the empirical mean m and covariance C of feasible pilot points; whiten z=C^{-1/2}(x-m). Initialize z_0 from a cheap feasible point, truncated Gaussian, or previous minibatch. For t=0,...,L-1, draw g~N(0,I_n), set u=g/(||g||_2+1e-8), compute every scalar interval constraint for z_t+s u, intersect the intervals to obtain [s_min,s_max], draw s~Uniform[s_min,s_max], and set z_{t+1}=z_t+s u. Return x_t=m+C^{1/2}z_t after burn-in and thin every q steps. 3. The paper supplies the chord transition and logarithmic warm-start mixing dependence; estimate m, C, chord endpoints, and effective warmness empirically. Use C+delta I for numerical stability and reject or repair points outside K. 4. First experiment: use a 32- or 64-dimensional VAE latent space on CIFAR-10, constraining augmentations to a convex polytope formed by latent coordinate bounds and linear classifier-margin constraints. Compare hit-and-run against rejection sampling and Gaussian random-walk Metropolis at equal generated samples and wall-clock cost. Measure constraint violations, pairwise diversity, effective sample size, and downstream classifier accuracy. Success is higher effective sample size and validation accuracy at equal sampler time, especially for strongly biased initialization and small target tolerance.