Adaptive data embedding for curved spaces.

Recent studies have demonstrated the significance of hyperbolic geometry in uncovering low-dimensional structure within complex hierarchical systems. We developed a Bayesian formulation of multi-dimensional scaling (MDS) for embedding data in hyperbolic spaces that allows for a principled determination of manifold parameters such as curvature and dimension. We show that only a small amount of data are needed to constrain the manifold, the optimization is robust against false minima, and the model is able to correctly discern between Hyperbolic and Euclidean data. Application of the method to COVID sequences revealed that viral evolution leaves the dimensionality of the space unchanged but produces a logarithmic increase in curvature, indicating a constant rate of information acquisition optimized under selective pressures. The algorithm also detected a contraction in curvature after the introduction of vaccines. The ability to discern subtle changes and structural shifts showcases the utility of this approach in understanding complex data dynamics.