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 $\hat b_t(x) = \nablaφ(x)^\topη_t$, where $η_t\in\R^P$ 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 training-free generation and model combination. We demonstrate the approach on financial time series, turbulence, and image generation.

翻译：我们在随机插值子框架内发展了一种核方法用于生成建模，以线性系统替代神经网络训练。生成随机微分方程的漂移项为 $\hat b_t(x) = \nablaφ(x)^\topη_t$，其中 $η_t\in\R^P$ 通过可从数据计算的 $P\times P$ 系统求解得到，且 $P$ 独立于数据维度 $d$。由于估计存在误差，扩散系数 $D_t$ 会影响样本质量；根据Girsanov定理得到的最优系数 $D_t^*$ 在 $t=0$ 处发散，但这并不构成障碍，我们开发了一种能无缝处理该问题的积分器。该框架兼容多种特征映射——散射变换、预训练生成模型等——实现了免训练生成与模型融合。我们在金融时间序列、湍流和图像生成任务上验证了该方法的有效性。