# Tree density estimation

arXiv: https://arxiv.org/abs/2111.11971 · HTML: https://mlsanity.com/paper/2111.11971/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 24 references that are in the map.
- First posted on arXiv 23 Nov 2021.

## Builds on (most cited first, 2 of 2)

- [Optimal rates of entropy estimation over Lipschitz balls](https://mlsanity.com/paper/1711.02141/) (2017)
- [Copula-based anomaly scoring and localization for large-scale, high-dimensional continuous data](https://mlsanity.com/paper/1912.02166/) (2019)

## Abstract (arXiv)

We study the problem of estimating the density f(boldsymbol x) of a random vector boldsymbol X in mathbb Rd. For a spanning tree T defined on the vertex set 1,dots ,d, the tree density fT is a product of bivariate conditional densities. An optimal spanning tree minimizes the Kullback-Leibler divergence between f and fT. From i.i.d. data we identify an optimal tree T^* and efficiently construct a tree density estimate fn such that, without any regularity conditions on the density f, one has limn→ ∞ int |fn(boldsymbol x)-fT^*(boldsymbol x)|dboldsymbol x=0 a.s. For Lipschitz f with bounded support, mathbb E left int |fn(boldsymbol x)-fT^*(boldsymbol x)|dboldsymbol xright=Obig(n-1/4big), a dimension-free rate.