Adaptive data embedding for curved spaces.

Area of Science:

• Complex Systems Science
• Computational Geometry
• Bioinformatics

Background:

• Hyperbolic geometry is increasingly recognized for its ability to model complex hierarchical systems.
• Existing methods for data embedding in hyperbolic spaces lack principled parameter estimation.
• Understanding evolutionary dynamics requires robust methods for analyzing complex biological sequence data.

Purpose of the Study:

• To develop a Bayesian multi-dimensional scaling (MDS) method for hyperbolic data embedding.
• To enable principled estimation of manifold parameters like curvature and dimension.
• To apply this method to analyze viral evolution dynamics, specifically in COVID-19 sequences.

Main Methods:

• Developed a Bayesian formulation of multi-dimensional scaling (MDS) for hyperbolic spaces.
• Incorporated principled determination of manifold parameters (curvature, dimension).
• Validated the model's robustness and ability to distinguish hyperbolic from Euclidean data.

Main Results:

• The Bayesian MDS model requires minimal data to constrain manifold parameters.
• Optimization proved robust against false minima, accurately distinguishing hyperbolic and Euclidean data.
• Analysis of COVID-19 sequences showed viral evolution increases hyperbolic curvature logarithmically without changing dimensionality.
• A contraction in curvature was observed post-vaccine introduction.

Conclusions:

• The developed Bayesian MDS approach effectively uncovers low-dimensional structures in complex systems.
• Hyperbolic geometry provides a powerful framework for analyzing viral evolution and response to interventions.
• This method offers utility in discerning subtle changes and structural shifts in dynamic complex data.