Making generative models stable with kernel smoothing

One-step generative models use drift velocities to transform noise into data, but standard displacement-based approaches aren't mathematically conservative (they don't form gradient fields). This work replaces that velocity with a kernel density estimator that smooths both data and model scores, guaranteeing the result is a valid gradient field. The authors prove finite-particle convergence bounds showing the method scales as N^{-1/(d+4)}, with explicit tracking of how bandwidth and dimension affect performance.