Generative Modeling via Kernelized Stochastic Interpolants

Abstract

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 ${\hat{b}}_t(x)=\nabla \ddot{I}•(x{)}^{\top }\hat{I}{\cdot }_t$, where $\hat{I}{\cdot }_t\in {\mathbb{R}}^P$ 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 generation and model combination without neural network training. We demonstrate the approach on financial time series, turbulence, and image generation.