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A Comparison of Polynomial-Based Tree Clustering Methods

Pengyu Liu, Mariel Vázquez, Nataša Jonoska

TL;DR

The paper tackles clustering of tree-structured data by employing the tree distinguishing polynomial $P(T,x,y)$ encoded as a coefficient matrix $C_T$, enabling distance-based analysis with several distance measures and two autoencoder-based clustering approaches. It compares six distances—$d_E$, $d_{\bar{E}}$, $d_M$, $d_{\bar{M}}$, $d_C$, and $d_{BC}$—within $k$-medoids, and evaluates two autoencoders with $k$-means on the latent space, using a dataset of $100$ sets of $300$ rooted binary trees generated by the beta-splitting model with $\beta\in\{-1.5,-1,0\}$. Results show that entry-level normalized distances $d_{\bar{E}}$, $d_{\bar{M}}$, and $d_C$ achieve the highest clustering accuracy (roughly $0.89$–$0.90$), outperforming unnormalized distances ($\approx 0.62$) and autoencoder baselines ($0.76$–$0.79$). The findings support prioritizing normalized distance measures for tree-polynomial clustering and point to future work on non-binary trees, larger datasets, non-coding RNA structures, and transformer-based methods for tree polynomials.

Abstract

Tree structures appear in many fields of the life sciences, including phylogenetics, developmental biology and nucleic acid structures. Trees can be used to represent RNA secondary structures, which directly relate to the function of non-coding RNAs. Recent developments in sequencing technology and artificial intelligence have yielded numerous biological data that can be represented with tree structures. This requires novel methods for tree structure data analytics. Tree polynomials provide a computationally efficient, interpretable and comprehensive way to encode tree structures as matrices, which are compatible with most data analytics tools. Machine learning methods based on the Canberra distance between tree polynomials have been introduced to analyze phylogenies and nucleic acid structures. In this paper, we compare the performance of different distances in tree clustering methods based on a tree distinguishing polynomial. We also implement two basic autoencoder models for clustering trees using the polynomial. We find that the distance based methods with entry-level normalized distances have the highest clustering accuracy among the compared methods.

A Comparison of Polynomial-Based Tree Clustering Methods

TL;DR

The paper tackles clustering of tree-structured data by employing the tree distinguishing polynomial encoded as a coefficient matrix , enabling distance-based analysis with several distance measures and two autoencoder-based clustering approaches. It compares six distances—, , , , , and —within -medoids, and evaluates two autoencoders with -means on the latent space, using a dataset of sets of rooted binary trees generated by the beta-splitting model with . Results show that entry-level normalized distances , , and achieve the highest clustering accuracy (roughly –), outperforming unnormalized distances () and autoencoder baselines (–). The findings support prioritizing normalized distance measures for tree-polynomial clustering and point to future work on non-binary trees, larger datasets, non-coding RNA structures, and transformer-based methods for tree polynomials.

Abstract

Tree structures appear in many fields of the life sciences, including phylogenetics, developmental biology and nucleic acid structures. Trees can be used to represent RNA secondary structures, which directly relate to the function of non-coding RNAs. Recent developments in sequencing technology and artificial intelligence have yielded numerous biological data that can be represented with tree structures. This requires novel methods for tree structure data analytics. Tree polynomials provide a computationally efficient, interpretable and comprehensive way to encode tree structures as matrices, which are compatible with most data analytics tools. Machine learning methods based on the Canberra distance between tree polynomials have been introduced to analyze phylogenies and nucleic acid structures. In this paper, we compare the performance of different distances in tree clustering methods based on a tree distinguishing polynomial. We also implement two basic autoencoder models for clustering trees using the polynomial. We find that the distance based methods with entry-level normalized distances have the highest clustering accuracy among the compared methods.
Paper Structure (18 sections, 7 equations, 1 figure)

This paper contains 18 sections, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Methods
  3. Random Tree Generation
  4. Tree Polynomial
  5. Polynomial-Based Tree Clustering Methods
  6. Tree Polynomial Distances with K-Medoids Clustering
  7. Euclidean distances
  8. Manhattan distances
  9. Canberra and Bray-Curtis distances
  10. Clustering Experiment and Accuracy
  11. Tree Polynomial Autoencoders with K-Means Clustering
  12. Linear Autoencoder Model
  13. Convolutional Autoencoder Model
  14. Clustering Experiment and Accuracy
  15. Results
  16. ...and 3 more sections

Figures (1)

  • Figure 1: This figure displays the distribution of the accuracy of 8 polynomial-based tree clustering methods. The distance-based methods are colored in blue, and the autoencoder-based methods are in red. Each dot in a violin plot represents the accuracy for one set of random trees. The bars in each violin plot shows the mean, maximum and minimum observed accuracy over the 100 sets of random trees.