A supervised deep learning method for nonparametric density estimation

Nonparametric density estimation is an unsupervised learning problem. In this work we propose a two-step procedure that casts the density estimation problem in the first step into a supervised regression prob-lem. The advantage is that we can afterwards apply supervised learning methods. Compared to the standard nonparametric regression setting, the proposed procedure creates, however, dependence among the training sam-ples. To derive statistical risk bounds, one can therefore not rely on the well-developed theory for i.i.d. data. To overcome this, we prove an oracle inequality for this specific form of data dependence. As an application, it is shown that under a compositional structure assumption on the underlying density, the proposed two-step method achieves convergence rates that are faster than the standard nonparametric rates. A simulation study illus-trates the finite sample performance.