  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.
