Tree density estimation

TL;DR

This paper introduces a method for estimating a multivariate density using optimal spanning trees, achieving consistent and dimension-free convergence rates without regularity assumptions.

Contribution

It proposes a novel tree-based density estimator that identifies an optimal spanning tree and guarantees almost sure convergence with a dimension-free rate.

Findings

Consistent density estimation using optimal spanning trees.

Dimension-free convergence rate of O(n^{-1/4}) for Lipschitz densities.

No regularity conditions required on the density.

Abstract

We study the problem of estimating the density $f(\boldsymbol x)$ of a random vector ${\boldsymbol X}$ in $\mathbb R^d$. For a spanning tree $T$ defined on the vertex set $\{1,\dots ,d\}$, the tree density $f_{T}$ is a product of bivariate conditional densities. An optimal spanning tree minimizes the Kullback-Leibler divergence between $f$ and $f_{T}$. From i.i.d. data we identify an optimal tree $T^*$ and efficiently construct a tree density estimate $f_n$ such that, without any regularity conditions on the density $f$, one has $\lim_{n\to \infty} \int |f_n(\boldsymbol x)-f_{T^*}(\boldsymbol x)|d\boldsymbol x=0$ a.s. For Lipschitz $f$ with bounded support, $\mathbb E \left\{ \int |f_n(\boldsymbol x)-f_{T^*}(\boldsymbol x)|d\boldsymbol x\right\}=O\big(n^{-1/4}\big)$, a dimension-free rate.