Neural-Kernelized Conditional Density Estimation

Conditional density estimation is a general framework for solving various problems in machine learning. Among existing methods, non-parametric and/or kernel-based methods are often difficult to use on large datasets, while methods based on neural networks usually make restrictive parametric assumptions on the probability densities. Here, we propose a novel method for estimating the conditional density based on score matching. In contrast to existing methods, we employ scalable neural networks, but do not make explicit parametric assumptions on densities. The key challenge in applying score matching to neural networks is computation of the first- and second-order derivatives of a model for the log-density. We tackle this challenge by developing a new neural-kernelized approach, which can be applied on large datasets with stochastic gradient descent, while the reproducing kernels allow for easy computation of the derivatives needed in score matching. We show that the neural-kernelized function approximator has universal approximation capability and that our method is consistent in conditional density estimation. We numerically demonstrate that our method is useful in high-dimensional conditional density estimation, and compares favourably with existing methods. Finally, we prove that the proposed method has interesting connections to two probabilistically principled frameworks of representation learning: Nonlinear sufficient dimension reduction and nonlinear independent component analysis.