Identifiability and Estimation in High-Dimensional Nonparametric Latent Structure Models

TL;DR

This paper advances understanding of high-dimensional nonparametric latent models by establishing new identifiability conditions and near-optimal estimation rates, showing that increased dimensionality can aid rather than hinder analysis.

Contribution

It introduces a generalized identifiability theorem and develops a recovery procedure, providing a unified framework and near-optimal bounds for high-dimensional nonparametric latent models.

Findings

Increased dimensionality and variable diversity improve identifiability.

Sample complexity scales polynomially with dimension, defying the curse of dimensionality.

Proposed a perturbation-based recovery method using simultaneous diagonalization.

Abstract

This paper studies the problems of identifiability and estimation in high-dimensional nonparametric latent structure models. We introduce an identifiability theorem that generalizes existing conditions, establishing a unified framework applicable to diverse statistical settings. Our results rigorously demonstrate how increased dimensionality, coupled with diversity in variables, inherently facilitates identifiability. For the estimation problem, we establish near-optimal minimax rate bounds for the high-dimensional nonparametric density estimation under latent structures with smooth marginals. Contrary to the conventional curse of dimensionality, our sample complexity scales only polynomially with the dimension. Additionally, we develop a perturbation theory for component recovery and propose a recovery procedure based on simultaneous diagonalization.