A Data-Augmented Contrastive Learning Approach to Nonparametric Density Estimation

In this paper, we introduce a data-augmented nonparametric noise contrastive estimation method to density estimation using deep neural networks. By leveraging the idea of contrastive learning, our density estimator exhibits efficiency with a one-step and simulation-free evaluation process, imposes no constraints on the neural network, and is shown to be consistent and asymptotically automatically normalized. A novel data augmentation procedure allows us to mitigate the influence of the choice of reference distribution on our method. Non-asymptotic upper bounds for the expected $L_{2}$-risk and the expected total variation distance have been established, which achieve minimax optimal rates. Moreover, our new method exhibits inherent adaptivity to low dimensional structures of data with a faster convergence rate under a compositional structure assumption. Numerical experiments show the competitiveness of our new method compared with the state-of-the-art nonparametric density estimation methods.