Convex-Projected Diffusion Sampler

Use a VE diffusion sampler first, because the denoiser-to-score relation is explicit. Let the network output $\widehat m_\theta(x,t)$, either by retraining the final head with denoising-score matching or by converting an existing score prediction using $\widehat m_\theta=x+\sigma_t^2\widehat s_\theta$. Choose a bounded closed convex set $K$: for normalized images, an elementwise box $[-1,1]^d$ is the cheapest baseline; for vector data use a ball or a tighter coordinate box estimated from the training set with a safety margin. At every reverse-time grid point, compute the denoiser, project it, convert it to a score, and pass that score to the existing reverse drift. Pseudocode: x = sample_prior(); for k=T,...,1: m_hat = net(x,t[k]); m_proj = project_K(m_hat); s = (m_proj-x)/sigma(t[k])**2; x = x + h*reverse_drift(x,t[k],s) + g(t[k])*sqrt(h)*randn(); return x. The mathematical guarantee is the projection inequality and the bounded-output property; estimate $K$, score calibration, and numerical effects empirically. First test on CIFAR-10 with a small DiT or U-Net, comparing unprojected Euler-Maruyama, projected Euler-Maruyama, and deterministic probability-flow sampling at identical networks and step counts. Track FID, precision/recall, maximum denoiser norm, fraction of trajectories exceeding a large state norm, endpoint $W_1$ against a reference fine-step sampler, and how these quantities change with step size. Success means substantially fewer rare large-norm trajectories and bounded endpoint moments, with no meaningful FID degradation; a stronger result is improved FID or Wasserstein error at coarse step counts.