Training-Free Generative Modeling via Kernelized Stochastic Interpolants

We develop a kernel method for generative modeling within the stochastic interpolant framework, replacing neural network training with linear systems. The drift of the generative SDE is ${\widehat{b}}_t(x)=\mathrm{\nabla }\phi (x{)}^{\mathrm{\top }}{\eta }_t$, where ${\eta }_t\in {\mathbb{R}}^P$ solves a $P\times P$ system computable from data, with $P$ independent of the data dimension $d$. Since estimates are inexact, the diffusion coefficient $D_t$ affects sample quality; the optimal $D_t^{\ast }$ from Girsanov diverges at $t=0$, but this poses no difficulty and we develop an integrator that handles it seamlessly. The framework accommodates diverse feature maps – scattering transforms, pretrained generative models etc. – enabling training-free generation and model combination. We demonstrate the approach on financial time series, turbulence, and image generation.