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 $b_t(x) =\nabla \phi(x)^\top\eta_t$, 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 $D_t^*$ 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 generation and model combination without neural network training. We demonstrate the approach on financial time series, turbulence, and image generation.