Density Estimation in Infinite Dimensional Exponential Families

TL;DR

This paper introduces a new density estimation method within infinite dimensional exponential families parametrized by RKHS functions, using Fisher divergence minimization to achieve consistent estimates and better performance than kernel methods, especially in high dimensions.

Contribution

The paper proposes a Fisher divergence-based estimator for densities in infinite dimensional exponential families, overcoming limitations of traditional MLE methods and providing theoretical convergence rates.

Findings

Estimator is consistent when the true density is in the model

Achieves a convergence rate of n^{- ext{min}rac{2}{3},rac{2eta+1}{2eta+2}}

Outperforms kernel density estimators in simulations, especially as dimension increases

Abstract

In this paper, we consider an infinite dimensional exponential family, $\mathcal{P}$ of probability densities, which are parametrized by functions in a reproducing kernel Hilbert space, $H$ and show it to be quite rich in the sense that a broad class of densities on $\mathbb{R}^d$ can be approximated arbitrarily well in Kullback-Leibler (KL) divergence by elements in $\mathcal{P}$. The main goal of the paper is to estimate an unknown density, $p_0$ through an element in $\mathcal{P}$. Standard techniques like maximum likelihood estimation (MLE) or pseudo MLE (based on the method of sieves), which are based on minimizing the KL divergence between $p_0$ and $\mathcal{P}$, do not yield practically useful estimators because of their inability to efficiently handle the log-partition function. Instead, we propose an estimator, $\hat{p}_n$ based on minimizing the \emph{Fisher divergence}, $J(p_0\Vert p)$ between $p_0$ and $p\in \mathcal{P}$, which involves solving a simple finite-dimensional linear system. When $p_0\in\mathcal{P}$, we show that the proposed estimator is consistent, and provide a convergence rate of $n^{-\min\left\{\frac{2}{3},\frac{2\beta+1}{2\beta+2}\right\}}$ in Fisher divergence under the smoothness assumption that $\log p_0\in\mathcal{R}(C^\beta)$ for some $\beta\ge 0$, where $C$ is a certain Hilbert-Schmidt operator on $H$ and $\mathcal{R}(C^\beta)$ denotes the image of $C^\beta$. We also investigate the misspecified case of $p_0\notin\mathcal{P}$ and show that $J(p_0\Vert\hat{p}_n)\rightarrow \inf_{p\in\mathcal{P}}J(p_0\Vert p)$ as $n\rightarrow\infty$, and provide a rate for this convergence under a similar smoothness condition as above. Through numerical simulations we demonstrate that the proposed estimator outperforms the non-parametric kernel density estimator, and that the advantage with the proposed estimator grows as $d$ increases.