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Rootlets Hierarchical Principal Component Analysis for Revealing Nested Dependencies in Hierarchical Data.

Korey P Wylie¹, Jason R Tregellas^1,2

¹Department of Psychiatry, University of Colorado School of Medicine, Anschutz Medical Campus, Anschutz Health Sciences Building, 1890 N Revere Ct, Aurora, CO 80045, USA.

Mathematics (Basel, Switzerland)

|September 25, 2025

Summary

This summary is machine-generated.

A new rootlets hierarchical principal component analysis (hPCA) method overcomes limitations of traditional hierarchical clustering analysis (HCA). This advanced technique accurately reconstructs data hierarchies without imposing artificial structures.

Keywords:

Riemannian geometry eigendecomposition hyperbolic manifold manifold learning multivariate statistics

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Area of Science:

Machine Learning
Data Analysis
Multivariate Statistics

Background:

Hierarchical clustering analysis (HCA) is a common unsupervised learning technique.
HCA has limitations, including imposing artificial hierarchies and using fixed two-way mergers.
These limitations can misrepresent the underlying structure of non-hierarchical data.

Purpose of the Study:

To introduce a novel rootlets hierarchical principal component analysis (hPCA) method.
To address the limitations of traditional HCA by enabling adaptive multiway mergers.
To visualize nested dependencies using Riemannian geometry and the Poincaré disk.

Main Methods:

Developed rootlets hPCA, extending hPCA with multivariate statistics for adaptive multiway mergers.
Employed Riemannian geometry for visualizing nested dependencies, projecting onto the Poincaré disk.
Algorithm decomposes similarity matrices using rotations from SO(k) and limits eigenvalues per merger.

Main Results:

Rootlets hPCA successfully constructs and merges nested clusters based on leading principal components.
The method limits the number of distinct eigenvalues for any merger.
Validated on simulated and neuroimaging datasets, accurately reconstructing known hierarchies.

Conclusions:

Rootlets hPCA offers an advanced alternative to HCA for hierarchical data analysis.
The method avoids imposing artificial hierarchies, providing more accurate data representation.
Its visualization technique effectively maps complex dependencies onto a hyperbolic manifold.