Dimension-independent rates for structured neural density estimation

TL;DR

This paper proves that deep neural networks can learn structured densities like images and text with convergence rates independent of ambient dimension, depending only on the structure's maximum clique size.

Contribution

It establishes dimension-independent convergence rates for neural density estimation based on the underlying Markov structure, providing theoretical support for deep learning's effectiveness in high-dimensional data.

Findings

Neural networks achieve rate $n^{-1/(4+r)}$ in density estimation.

Optimal $L^1$ rate is $n^{-1/(2+r)}$, depending on clique size.

Rates are independent of ambient data dimension.

Abstract

We show that deep neural networks achieve dimension-independent rates of convergence for learning structured densities such as those arising in image, audio, video, and text applications. More precisely, we demonstrate that neural networks with a simple $L^2$-minimizing loss achieve a rate of $n^{-1/(4+r)}$ in nonparametric density estimation when the underlying density is Markov to a graph whose maximum clique size is at most $r$, and we provide evidence that in the aforementioned applications, this size is typically constant, i.e., $r=O(1)$. We then establish that the optimal rate in $L^1$ is $n^{-1/(2+r)}$ which, compared to the standard nonparametric rate of $n^{-1/(2+d)}$, reveals that the effective dimension of such problems is the size of the largest clique in the Markov random field. These rates are independent of the data's ambient dimension, making them applicable to realistic models of image, sound, video, and text data. Our results provide a novel justification for deep learning's ability to circumvent the curse of dimensionality, demonstrating dimension-independent convergence rates in these contexts.