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Consistency of Nonparametric Density Estimators in CAT(0) Orthant Space

Yuki Takazawa, Tomonari Sei

TL;DR

This work establishes the statistical consistency of two nonparametric density estimators in CAT(0) orthant spaces, including BHV tree space, by extending log-concave projection theory and proving its continuity, and by introducing a boundary-bias–corrected kernel density estimator with uniform consistency. It shows that a log-concave projection exists and is unique (nu-almost everywhere) under mild moment conditions, and proves the continuity of the projection map with respect to Wasserstein-1 convergence, enabling consistency results for the log-concave MLE. For kernel density estimation, the authors diagnose boundary bias in the standard KDE on orthant spaces and propose a modified kernel, proving uniform consistency under standard bandwidth assumptions and extending the results to product spaces. The findings advance nonparametric inference on phylogenetic-tree data, offering theoretically sound tools for density estimation in complex geometric spaces and highlighting avenues for computational and methodological extensions in high dimensions.

Abstract

The inference of evolutionary histories is a central problem in evolutionary biology. The analysis of a sample of phylogenetic trees can be conducted in Billera-Holmes-Vogtmann tree space, which is a CAT(0) metric space of phylogenetic trees. The globally non-positively curved (CAT(0)) property of this space enables the extension of various statistical techniques. In the problem of nonparametric density estimation, two primary methods, kernel density estimation and log-concave maximum likelihood estimation, have been proposed, yet their theoretical properties remain largely unexplored. In this paper, we address this gap by proving the consistency of these estimators in a more general setting$\unicode{x2014}$CAT(0) orthant spaces, which include BHV tree space. We extend log-concave approximation techniques to this setting and establish consistency via the continuity of the log-concave projection map. We also modify the kernel density estimator to correct boundary bias and establish uniform consistency using empirical process theory.

Consistency of Nonparametric Density Estimators in CAT(0) Orthant Space

TL;DR

This work establishes the statistical consistency of two nonparametric density estimators in CAT(0) orthant spaces, including BHV tree space, by extending log-concave projection theory and proving its continuity, and by introducing a boundary-bias–corrected kernel density estimator with uniform consistency. It shows that a log-concave projection exists and is unique (nu-almost everywhere) under mild moment conditions, and proves the continuity of the projection map with respect to Wasserstein-1 convergence, enabling consistency results for the log-concave MLE. For kernel density estimation, the authors diagnose boundary bias in the standard KDE on orthant spaces and propose a modified kernel, proving uniform consistency under standard bandwidth assumptions and extending the results to product spaces. The findings advance nonparametric inference on phylogenetic-tree data, offering theoretically sound tools for density estimation in complex geometric spaces and highlighting avenues for computational and methodological extensions in high dimensions.

Abstract

The inference of evolutionary histories is a central problem in evolutionary biology. The analysis of a sample of phylogenetic trees can be conducted in Billera-Holmes-Vogtmann tree space, which is a CAT(0) metric space of phylogenetic trees. The globally non-positively curved (CAT(0)) property of this space enables the extension of various statistical techniques. In the problem of nonparametric density estimation, two primary methods, kernel density estimation and log-concave maximum likelihood estimation, have been proposed, yet their theoretical properties remain largely unexplored. In this paper, we address this gap by proving the consistency of these estimators in a more general settingCAT(0) orthant spaces, which include BHV tree space. We extend log-concave approximation techniques to this setting and establish consistency via the continuity of the log-concave projection map. We also modify the kernel density estimator to correct boundary bias and establish uniform consistency using empirical process theory.
Paper Structure (31 sections, 25 theorems, 151 equations, 3 figures)

This paper contains 31 sections, 25 theorems, 151 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. CAT(0) orthant space
  4. Orthant space
  5. CAT(0) space
  6. CAT(0) orthant space
  7. Log-concave density on CAT(0) orthant space
  8. Kernel density estimator
  9. Theoretical Properties of Log-concave Projection
  10. Log-concave projections
  11. The uniqueness and existence of log-concave projections
  12. Continuity of log-concave projection map
  13. Computation and visualization of the log-concave MLE
  14. Consistency of Kernel Density Estimator
  15. Kernel density estimator and boundary bias
  16. ...and 16 more sections

Key Result

Lemma 2.1

Let $(\mathcal{H}, d)$ be a Hadamard space, and let $\gamma, \gamma^\prime : [0,1] \to \mathcal{H}$ be two arbitrary geodesics. Then, the distance function satisfies the following joint convexity inequality:

Figures (3)

  • Figure 1: (Left): 3-spider. (Center): $\Omega^{(4)}$ corresponding to the phylogenetic tree space $\mathcal{T}_4= \mathcal{O}(\mathcal{E}^{(4)}, \Omega^{(4)})$. Each vertex of the labeled graph corresponds to a singleton set in $\Omega$, and the set of two vertices connected by a line segment corresponds to a two-element set in $\Omega$. (Right): A part of $\mathcal{T}_4$.
  • Figure 2: Log-concave MLE in the 3-spider space. Each subfigure displays the three orthants, overlaying the true density (black dotted) and the log-concave MLE (green solid). Panel (a) corresponds to the Gaussian-type density $f_1$, and panel (b) to the mixture density $f_2$.
  • Figure 3: Kernel density estimation in the 3-spider space. Each subfigure shows the three orthants with the true density (black dotted), the original estimator with kernel $K_1$ (blue dashed), and the modified estimator with kernel $K_2$ (orange solid). Panel (a) corresponds to $f_1$, and panel (b) to $f_2$.

Theorems & Definitions (52)

  • Example 1
  • Example 2: BHV Treespace
  • Lemma 2.1: Bacak2014-nn
  • Theorem 2.2: Gromov1987-vnMiller2015-jm
  • Example 3
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 42 more