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)=∇φ(x)⊤ηt, where ηt∈RP solves a P×P system computable from data, with P independent of the data dimension d. Since estimates are inexact, the diffusion coefficient Dt affects sample quality; the optimal Dt∗ 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.
We propose a framework for generative modeling of continuous-time processes from irregularly and asynchronously recorded data. It is based on the matching of generators and accommodates discontinuous trajectories. Analytical formulas for diffusion and jump bridges yield a family of reference generators that a neural network is trained to match. The key ingredient is that, for our constructed jump bridge, a parametrization of the jump kernel densities by scaled Gaussians admits closed-form expressions for the Kullback-Leibler divergence, allowing simulation-free training.
This paper provides a theoretical account of memorization in stochastic interpolation models. By leveraging closed-form expressions for the optimal velocity field and the associated score function, we show that, in the continuous-time oracle setting, both deterministic and stochastic generation processes recover training samples. Under Euler discretization, generated samples remain centered around training samples, with deviations controlled by the step size. We further analyze generation in the presence of estimation errors and show that accumulated estimation errors control the endpoint deviation from the training set. These results imply that the generated sample admits a representation as a training sample perturbed by three controlled terms: a discretization-induced bound, an estimation-error-induced bound, and stochastic Gaussian noise. Based on this characterization, we provide theoretical definitions of overfitting and underfitting in generative models. Synthetic simulations support our theoretical findings.
We propose kernel-gradient drifting, a one-step generative modeling framework that replaces the fixed Euclidean displacement direction in drifting models with directions induced by the kernel itself. Standard drifting is attractive because it enables fast, high-quality generation without distilling a large pretrained diffusion model, but its theory is currently understood mainly for Gaussian kernels, where the drift coincides with smoothed score matching and is identifiable. Our gradient-based reformulation exposes this score-based structure for general kernels: the resulting drift is the score difference between kernel-smoothed data and model distributions, yielding identifiability for characteristic kernels and a smoothed-KL descent interpretation of the drifting dynamics. Since kernel gradients are intrinsic tangent vectors, the same construction extends naturally to Riemannian manifolds and to discrete data via the Fisher-Rao geometry of the probability simplex. Across spherical geospatial data, promoter DNA and molecule generation, kernel-gradient drifting enables state-of-the-art one-step generation beyond the Euclidean setting without distillation.
Maria Esteban-Casadevall, Jorge Carrasco-Pollo, Max Welling +3