Ergodic and foliated kernel-differentiation method for linear responses of random systems

We extend the kernel-differentiation method for the linear response (parameter-derivative of averaged observables) of random dynamical systems. First, for the linear response of physical (or stationary) measures, we extend the method to an ergodic version, which is sampled by an infinitely long sample path, so it is more efficient than previous results. This is achieved by combining the likelihood ratio trick, decay of correlations, and ergodic theorem. Second, when the noise and perturbation are along a given foliation, we show that the method is still valid for both finite and infinite time. These results are derived using basic calculus via a microscopic view of transfer operators. We use the ergodic formula to numerically compute the linear response of a tent map with additive noise. We use the foliated formula to compute the linear response of an unstable neural network with 51 layers $\times$ 9 neurons, with respect to the bias parameter. We show that adding foliated noise incurs a smaller error than adding noise in all directions, in terms of approximating the trend between the averaged observable and the parameter. We give a rough error analysis of the kernel-differentiation method and show that it can be expensive for small noise. Finally, we derive the three basic linear response methods (path-perturbation, divergence, and kernel-differentiation methods) in simplified settings, and propose a potential future program unifying them.