A Spectral Approach to Gradient Estimation for Implicit Distributions
This addresses a bottleneck in learning and inference with implicit distributions, offering a principled solution for out-of-sample gradient estimation, though it appears incremental as it builds on existing spectral and kernel methods.
The paper tackles the problem of gradient estimation for implicit distributions (without tractable densities) by developing a method based on Stein's identity and spectral decomposition, which directly estimates the gradient function for out-of-sample extension. It demonstrates effectiveness in applications like gradient-free Hamiltonian Monte Carlo and variational inference.
Recently there have been increasing interests in learning and inference with implicit distributions (i.e., distributions without tractable densities). To this end, we develop a gradient estimator for implicit distributions based on Stein's identity and a spectral decomposition of kernel operators, where the eigenfunctions are approximated by the Nyström method. Unlike the previous works that only provide estimates at the sample points, our approach directly estimates the gradient function, thus allows for a simple and principled out-of-sample extension. We provide theoretical results on the error bound of the estimator and discuss the bias-variance tradeoff in practice. The effectiveness of our method is demonstrated by applications to gradient-free Hamiltonian Monte Carlo and variational inference with implicit distributions. Finally, we discuss the intuition behind the estimator by drawing connections between the Nyström method and kernel PCA, which indicates that the estimator can automatically adapt to the geometry of the underlying distribution.