Divergence-kernel method for linear responses of densities and generative models

We derive the divergence-kernel formula for the linear response of random dynamical systems. Specifically, the pathwise expression is for the parameter-derivative of the marginal or stationary density, not an averaged observable. Our formula works for multiplicative and parameterized noise over any period of time; it does not require hyperbolicity. Then we derive a Monte-Carlo algorithm for linear responses. We develop a new framework of generative models, DK-SDE, where the model is a parameterized SDE, that (1) directly uses the KL divergence between the empirical data distribution and the marginal density of the SDE as the training objective, and (2) accommodates parametrizations in both drift and diffusion over a long time span, allowing prior structural knowledge to be incorporated explicitly. The optimization is done by gradient-descent enabled by the divergence-kernel method, which involves only forward processes and therefore substantially reduces memory cost. We demonstrate the new model on a 20-dimensional Lorenz system.