Nonparametric estimation of a factorizable density using diffusion models
To address the curse of dimensionality in high-dimensional nonparametric density estimation, this paper considers densities exhibiting a low-dimensional factorized structure—i.e., statistical independence across variable groups. We propose diffusion models as implicit density estimators and, for the first time within a statistical framework, establish that under the factorization assumption, the resulting estimator achieves a dimension-free minimax-optimal convergence rate in total variation distance—thereby circumventing the curse of dimensionality. To explicitly encode structural priors, we design a sparse weight-sharing neural network architecture that adaptively models low-dimensional components. Theoretical analysis confirms the improved statistical efficiency, while empirical results demonstrate superior estimation accuracy and enhanced interpretability in high-dimensional sparse settings—all without sacrificing the flexibility inherent to nonparametric methods.