Factor-Assisted Density Estimation: When Ambient Dimension Enables Oracle Rates

Abstract

We study density estimation in the high-dimensional factor model, whose fixed-dimensional latent factors have an unknown smooth density. We propose factor-assisted density estimation (FADE), a non-iterative procedure that applies kernel density estimation to spectral estimates of the latent scores. The score proxies are dependent and do not follow the latent density, so standard kernel-density arguments do not apply. A refined rowwise expansion separates error into a conditionally centered Gaussian perturbation and a uniformly controlled remainder. We show that FADE achieves the oracle rate in sufficiently high ambient dimension and prove a minimax lower bound separating ordinary nonparametric difficulty from a Gaussian deconvolution obstruction. Together, the upper and lower bounds establish rate optimality of FADE in specified high-dimensional regimes. We also derive $L^1$ upper and lower bounds for the induced observation density, distinguishing latent-distribution uncertainty from the cost of estimating a growing number of parameters. These results formalize a blessing of ambient dimension: sufficiently many observed coordinates can make a simple spectral reduction statistically lossless even for the nonparametric task of recovering an unobserved low-dimensional density.