Optimal Phylogenetic Reconstruction from Sampled Quartets

2026-04-19

AI Summary

This work addresses the problem of efficiently reconstructing an unknown phylogenetic tree and predicting unseen quartets from only Θ(n) randomly labeled noisy quartet samples. The authors propose an algorithm based on quartet embedding and detection, integrating semidefinite programming with PAC learning theory to achieve (1−ε)-approximate recovery of the true tree’s quartet distances at a sample complexity near the information-theoretic lower bound. Their main contributions include the first method to attain near-optimal tree reconstruction performance with only Θ(n) samples, along with establishing a tight bound of Θ(n) on the Natarajan dimension of the class of phylogenetic trees—significantly improving upon existing approaches that rely on dense or substantially larger sample sets.

Technology Category

• Machine Learning: Semi-Supervised Learning
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Quartet Reconstruction, the task of recovering a phylogenetic tree from smaller trees on four species called \textit{quartets}, is a well-studied problem in theoretical computer science with far-reaching connections to statistics, graph theory and biology. Given a random sample containing $m$ noisy quartets, labeled by an unknown ground-truth tree $T$ on $n$ taxa, we want to output a tree $\widehat T$ that is \textit{close} to $T$ in terms of quartet distance and can predict unseen quartets. Unfortunately, the empirical risk minimizer corresponds to the $\mathsf{NP}$-hard problem of finding a tree that maximizes agreements with the sampled quartets, and earlier works in approximation algorithms gave $(1-\eps)$-approximation schemes (PTAS) for dense instances with $m=Θ(n^4)$ quartets, or for $m=Θ(n^2\log n)$ quartets \textit{randomly} sampled from $T$. Prior to our work, it was unknown how many samples are information-theoretically required to learn the tree, and whether there is an efficient reconstruction algorithm. We present optimal results for reconstructing an unknown phylogenetic tree $T$ from a random sample of $m=Θ(n)$ quartets, corrupted under the Random Classification Noise (RCN) model. This matches the $Ω(n)$ lower bound required for any meaningful tree reconstruction. Our contribution is twofold: first, we give a tree reconstruction algorithm that, not only achieves a $(1-\eps)$-approximation, but most importantly \textit{recovers} a tree close to $T$ in quartet distance; second, we show a new $Θ(n)$ bound on the Natarajan dimension of phylogenies (an analog of VC dimension in multiclass classification). Our analysis relies on a new \textit{Quartet-based Embedding and Detection} procedure that identifies and removes well-clustered subtrees from the (unknown) ground-truth $T$ via semidefinite programming.

Problem

• Phylogenetic Reconstruction
• Quartet Reconstruction
• Sample Complexity
• Random Classification Noise
• Systematics

Innovation

• Quartet Reconstruction
• Random Classification Noise
• Natarajan Dimension
• Semidefinite Programming
• Phylogenetic Tree Learning